Positive curvature and beyond: A status report and future peek
Grove, Karsten
We will provide an overview of some of the main results around manifolds with positive and non-negative sectional curvature and discuss established and evolving approaches to the area.
 Well first of all I'd like to express how honored I am to be part of this celebration of Marcel Berger's life and work. So he had a tremendous impact on my own work, the support he provided, support and interest, etc. meant a lot to me. And like Tom said, we met in the early 70s, I met Berger in the early 70s and a whole group of his students. I don't know if all of them were students but that was my sense. Since then I found out that I looked up students on the math signet and also on the, that's right. And so it didn't look like all the ones I knew. So I thought I knew many more and turns out that I think it's incomplete. So, but independent of that I feel that, so he always had a large nice group of students and from day one I felt like, oh these are so nice to be around and it's nice to be around Berger. And their work and the whole attitude and support really meant a lot to me. In a certain sense I feel like an orphan in Orhus because I grew up with topologists at the time. They were very, very algebraic topologists. So I broke from that and I wanted to do geometry. And so Marcel Berger was really my idol and then clinging back to the big main figures in Europe and the world actually at that point. So I thought I would talk about, I took also the title of the conference literary. So I wanted to give like a survey, talk about past, present and future. On one of the topics where Marcel made a tremendous impact, namely manifold to positive curvature. So let me tell you immediately. So I want to focus on, so I'll write this as M plus. So manifolds in M plus are simply manifolds with sexual curvature positive. But here comes the beyond part. So the beyond part is I cannot help also to talk about manifolds of non-negative sexual curvature. And I may even go a little bit further because they're very natural classes related to one another. Namely those that are called manifold of almost non-negative curvature. For now, I'll just write it like this and I'll explain what this means in a little bit. So this of course is really where Berger has tremendous impact. And much of my own work has been concerned with this area. But I think I'll focus as much as I can on generalities in this talk. So I will focus general results as much as I can. And then things stop quickly actually. But still, now, so let me just remind you geometrically what this means. So one way to think about this is simply the following. If you take two vectors at a point P and you look at the corresponding geodesics, as long as they are minimal, the curvature positive means that the distance between the endpoints is smaller than the distance between the endpoints in the tangent space. And greater equal just means that they are less than or equal to. So this is the Hinsch version of Topologov's theorem. There are many equivalent formulations of that theorem. And all of this started with these comparisons theorems, comparisons of Jacobi fields. The Rauch-1 theorem, it starts out comparison of Jacobi fields that start out by being zero at a point. And then there's Rauch-2. I don't know why it's called Rauch-2 because it's due to Berger. So this is about Jacobi fields that start off by having some value, but covariant derivative zero at the point. And those two are really basic results that led to the proof of Topologov's theorem. So now, okay, so I think I will use this board. No? The reason I'm doing this is I thought I had figured it all out to avoid these shadows, but it didn't quite work. At any rate, let me start with the most general results in this, if I take the area as a whole, okay? So there are two, I don't think anyone gets offended. There are two milestone results of general type. One of them is due to Chigur and Gromal. It's a Sol theorem. Originally, Wolfgang Meier did some previous work with Gromal also. So it simply says the following. If m is a manifold with non-negative curvature, I assume they're complete all of them because otherwise we know for Gromal there's no theorems, right? So m is always complete but non-compact. This implies that m is actually the total space of a compact manifold that sits in isometrically. So m, so there exists a compact sub-manifold and this is totally convex, meaning that any geodesic starting and ending at s is in s. So in particular, it's really metrically embedded in here, totally geodesic, such that m is diffeomorphic to the normal bundle. So the total space of the normal bundle and this equivalence is up to diffeomorphism. All right, so this is one theorem. This is, you can think of it as, I mean, I like to think of it as a structure theorem or reduction theorem if you like. It's a structure theorem because it tells you what such manifolds look like, what's the structure of these manifolds. So they're all vector bundles over something compact from the same category. So it follows here that of course s is also in the same class and I should add that this is what's called the sole, of course. If m happens to be in this class, if it happens to have positive curvature, then the sole is a point. And in other words, then m is just diffeomorphic to rn, which means that I won't talk about it anymore. So we have a complete classification of positively curved manifolds that are non-compact in the sense of the diffeomorphism type. So in that sense, it's also a reduction theorem because now we're reduced to think about compact manifolds. So for the rest of the talk, I'll talk about compact manifolds in these classes. But before I move on to this, there's an obvious set of problems related to this, namely the converse problem. So you can imagine, given so let's see how this is working. Maybe it's not working at all. Oh, no, it's not working. Only if I put it all the way up. No, this is not going to work. I thought I had these brilliant ideas. You see, they teach us things that I never learned in any other place but Notre Dame. If there are two blackboards, they teach you from day one, I think, that you should always start with the one that's behind because of the shadow issue. I can't even see it now, so maybe I think I'll just give up and you will have to live with that shadow. Okay, so I'll pull it down again because this becomes much more comfortable for me, but now I started with the back button. God. All right, I'll just say it in words, okay? I'll just move on to something else. So what's the converse? The converse problem is you give yourself a compact manifold of non-negative curvature and a vector bundle over it. The question is, can you put a complete metric of non-negative curvature on the total space? Okay, now this is known to be false in general. There are lots of examples, the first were due to Waldschap and Oseiden and many more since then due to Bellegrad and Kapovitz. All of those examples are examples with infinite fundamental group. No counter example is known for finite, in particular, simply connected manifolds. Now, so and the original question by Chig and Grimal was if what about bundles over spheres? Even that is not known, only in dimensions up to the base being five-dimensional. However, Rieger's a long time ago in his thesis proved that over spheres, stably it's true. Okay, so you can add a large enough trivial bundle and then it's true. Okay, you can put a metric of non-negative curvature on it and recently it's been a lot of work by David Gonzalez about all other kinds of manifolds where you stable it's true. In particular, I think by now it's known to take any known positive recurve manifold then it's true all vector bundles over those if you add a trivial bundle. Okay, so that's a reasonable general question. Okay. All right, so this was the converse of that. So now let me pull this board down again and write down the other sort of general result. Oh, I can't do that now, right? Because I'm not tall enough. Yeah. Yeah. This is taking more time than I planned. When I came I saw all these boards I said, oh, I can finish this talk in no time. Okay. Yeah, I really messed up with this here. All right, so what's the other one? You probably guessed it. So this is the Benin-Uber theorem. It's a theorem due to Gromov and it simply says so like I said from now on we focus on compact manifolds. So one version of it is the following. If mn, so it exists a constant depending on the dimension n such that if m is in this class of non-negative curvature then the sum or the dimension of the homology of m, any field of coefficients is bounded by this constant n. Okay. Now there's a more general version of this theorem that doesn't care about what the lower curvature bound is and I should also add that the same thing is true for the fundamental group, pi1 of m is generated by less than some constant also. This one is a very simple beautiful consequence of Tobinogov's theorem. This one is really hard. It's a deep theorem and it's sort of the starting point is actually a rough version of this theorem that says that the topology so but it was proved only after the sole theorem but if you look at a complete manifold of non-negative curvature then it's a compact type. So that was Gromov's observation meaning that if you go far enough away there are no critical points for the distance function to the sole. So that means that it has the type of a compact manifold with boundary. That really is one of the key starting points in his proof of this thing for compact manifolds. Okay. But I won't say anything more about it. So now the more general result is for any curvature bound and then a bound also on the diameter. Of course in non-negative curvature you don't see that because you can always scale it. You can even go as far as you could say whatever we're going to discuss from now on we scale the diameter to be some number. Okay. Maybe pi. We like pi for the unit sphere maybe or else and then it's just a question about what is the lower curvature with the minimum of the lower bound for the curvature. Right. That's like a one function of this class of manifolds and this theorem says that if that function is greater than equal to zero then this is true but there's a value for all of them. Okay. There's a value that depends on this number. Okay. However that value is really large and so in particular if we go back to this general situation it does not distinguish these classes. Okay. You can you could say if you look at this thing this it's kind of the same number. There's no at this point. Okay. So one of the huge problems I would say that is is that we don't even know that these classes are different while we do at the level of fundamental groups. Okay. So at the level of fundamental groups these are really distinct classes. Here for example we have Born-Emeyer's theorem. The classical theorem. So now this is positive curvature. That's a general theorem of positive curvature. So m in m plus implies pi one of m is finite. This is the classical Born-Emeyer's theorem. It's true for retrocurvature even. And of course this is not true for the other classes. Okay. But so but actually you know what the structure is. You know a lot about the structure manifolds in these other classes. So for example in the class m zero you know that any compact manifold has a finite cover which is at least diffeomorphic to a simply connected one with non-negative terms across the torus. Okay. And what is this last class? Maybe I now should tell you what this means. So what does it mean that m is in this class? Well it means a couple of ways to say it. One is to say that there exists a family of metrics. Can you read this by the way up there? Well that's impressive because my handwriting is bad too. But I'll try to say it out loud. So there's a family of metrics on m so that the section curvature of m relative to this metric is greater than or equal to minus one. And the diameter of m with this metric approaches zero. That's one way of saying it. So this means collapse with lower curvature bound. Okay. This is equivalent by rescaling. You can see that there exists g i. Here's another way of saying it. So the sexual curvature of m i m g i is greater than or equal to let's say minus one divided by i where i goes to infinity. So this approach is zero getting closer and closer to zero. And let's say the diameter of m g i is a fixed number maybe pi. Okay. So this is why we call them almost non-negative and correct of course. And many folks in this class have not been studied that much but I think they really are important for understanding the whole picture and for understanding in general manifolds with lower curvature bound. Okay. Just like in the case of manifolds with bounded curvature, where almost flat manifolds play a central role in the work of Chica Foucaille and Gromov. Okay. So that's expected but hasn't been really developed. All right. By the way in this class any manifold in this class is also finally covered by a compact manifold that fibers over a nil manifold with fiber being simply connected. Okay. But in general the class of fundamental group is larger. But if we move on to assume that m is simply connected then there's not a single example known that of even going all the way to almost non-negative curvature that couldn't possibly have positive curvature. No obstructions are known. Okay. Let me add so this is so what are the other general theorems about positive curvature? They also go back a long time. They go back to synths or sing. So it's also about just about the fundamental group. If m is even dimensional this implies that pi1 is either trivial or z mod 2. This is when m is orientable and this is of course when it's not. Okay. And if it is odd dimensional then m is orientable. And these are results going back to synth in the 30s. Okay. And all of them are sort of like baby Morse theory in the sense. You look at the minimum general length of general minimal shortest closed deodesic or whatever in the homotopic class and you show that it has to be trivial. Okay. So this is just a simple variation formula. So in this example here you take a suppose m is not simply connected. You take a shortest closed deodesic that's non-trivial. You find from the assumptions a parallel field and you just look at the variation in that direction and then you find shorter curves. Okay. There are several theorems like this and I actually all put them in the same block. I kind them synth type theorems. Okay. There's a theorem due to Frankel that says in positive curvature if you have two totally deodesic sub manifolds they intersect if their dimensions are too big. It's exactly the same proof kind of going back to Weinstein if you have an isometry and such a manifold you find fixed points in even the mentions if it preserves orientation. And that by the way is very similar to Berger's theorem about zeros of killing fields. Okay. So he proved that any killing field on an even dimensional manifold has a zero and in order mentions it's true that if you have two killing fields they're dependent at one point. Okay. So these are the general results that I know of for manifolds of positive curvature without any further assumptions. Okay. So that's end of story in that sense. Okay. There's much more work that has been done but all of it with additional assumptions. So for example what what Marcel did of course was to prove the quarter pinching theorem and he proved other things very interesting things positive curvature. But these are not like completely general results. Okay. So I'm trying to stick with completely general results at this point and see what what else one can do. All right. So now I want to move on to some basic conjectures. And this is really not working well. Okay. Okay. Can you see it? Okay. I know the theorem so I don't need to see it. Okay. I can't see it. All right. So what are some main conjectures? Maybe I'll give up and just put this one here and then then do it reverse afterwards. Okay. So we can't avoid Hopf and there was mentioned by Bruce earlier today. So the Hopf has two basic conjectures attributed to Heinz-Hopf. One of them is that the Euler characteristic of M is greater than or equal to zero when M is non-negatively curved and actually it should be positive when M has positive sectional curvature. And the other one, the one that Bruce mentioned is the question whether S2 cross S2 is in or not. I guess if it's a conjecture, the conjecture was if it's not, but maybe Hopf asked just asked the question. So, okay. Now this has two very natural more general related questions I would say. So one is the following. If you take two manifolds N and V and take their product, there is not a single such example known to have positive curvature. I'm not saying conduct time anything. I'm just saying there's not such an example known yet. So of course, here's what we're talking about as 2 cross S2. Well, I mean this is not, if I take them to be simply connected, okay, take them to be simply connected. So only Rp2 cross Rp2 doesn't have positive curvature. So the zoom N and V is simply connected. And the other direction is how about you could think about symmetric spaces and I'll use, I guess this goes back to best this notation maybe, I think so, right. Compact rank one symmetric space, okay. So S2 cross S2 has ranked two, okay. And you could ask this question for any symmetric space a rank at least two. Is there a not right. Plus I'm saying it shouldn't have positive curvature. Oh, yes. Oh, then I can't attribute this. I just wanted to do that. Symmetric space, darn. Symmetric space, thank you. Of rank greater than or equal to two. Yes, thank you. Of course, that's what I meant. Yes, the other ones do. All right, so these are basic questions and here's a basic conjecture due to bot. That says that the, if we look at the, if M is also in this class, then the sequence of bedding numbers of the loop space of M grows at most polynomially. Okay. There's, there's a, this is, I'm not sure this is the formulation bot had because it's not in any, it's not written down, but bot wrote a paper with the Berger way where they looked at pinched manifold. So in this class with two bounds of the curvature and they provided estimates for the bedding numbers, but they're way from this way off from there and the exponential growth. They are estimates in terms of the pinching, but they have nothing to do with this conclusion that you would like. Okay. Now, if you focus, so you could also, another question is, are you thinking of any field like in Gromov's theorem or what not? Okay. But if you, if you, so here we could say F and this could be any field, maybe. That's a reasonable conjecture, but it becomes really obviously interesting in particular. There's a lot of consequences. If you take F to be the rational numbers, because then this condition is actually equivalent to the following that the dimension of the homotropic groups, the rational homotropic groups is finite. Okay. So this follows via the minimal model theory of Solomon and so on and so forth. This is old also, right? This is also way past. Okay. Now, this by itself, I'll just write down, it has lots of consequences that have been proved, topologically. One of them is that the Euler characteristic is then greater than or equal to zero. This follows from this. Okay. So, so the butt conjecture would imply one part of the Huff conjecture at least. The other part is that it's, and so, so which suggests what to try to do if you want to prove the Huff conjecture, if you believe butts. So, because being positive for the Euler characteristic is then equivalent to all odd rational better numbers to be zero. Okay. So in a sense, you might as well just prove that to prove the Huff conjecture. All right. So, all right. Another consequence is related to Gromov's theorem, and it is that the sum of better numbers, rational better numbers here, is at most two to the end where n is the dimension. See, that's a good bound, right? And in general, to distinguish these classes, one might want to prove just bounds that would allow you to distinguish them. That's what maybe one possibility, but that's, that's really all I wanted to mention here. Okay. Now, so, given this bleak situation about general results, I'd like to provide some other bleak news about how many examples do we know of this? Okay. So examples, and constructions. So what is, what are, how do we construct examples? Well, the, so let me talk about some constructions first. So if you start with a manifold m in m zero, and we look at, at, I'll call these projections. So you look at, at, at a, like a Croson map, m to n pi, where this is a Riemannian submersion, then this implies that this guy is also in this class, and you might be clever or lucky or whatever, and show that sometimes it is in this class. Okay. That's a very important construction. Now, if you look at, if you look at this class, Tom and I went out together last time, I'm losing my wise two now. If you look at this class, then you can take products, right? Products in this class work, of course. And then there's the bundles, and I would say in this class, what do I mean by bundles? It's the, that's the converse of this one in a certain sense. This is a very hard direction to go in. So basically, you start with something, you start with, with something, a base in a class, and you have a bundle over it, maybe a principal bundle, maybe a principal G bundle. I'll just stick to that right now. Then it follows that when B is in this class, that P is also in this class. So this is a way to construct many examples of almost non-negative curve manifolds that doesn't work in either of these categories in general, right? That's why this inverse, the converse of the sole theorem is very hard, right? So to go up to bundles is very difficult, right, to construct metrics. So these are some of the constructions. There's one other construction, and this is called the special gluings. And Jeff was the first to make such interesting constructions, namely, for example, and now, now I can use my word cross, okay? So if you take m and n, okay, compact rank one symmetric spaces, then Cheeg approved that their connected sum has a metric of non-negative curvature, okay? Basically using the, if you cut out a point from one of these space, you have a bundle and you change it so that it becomes product near the, near the boundary so that you can glue them together isometrically with a, by means of unit sphere. And there are other constructions about gluings that have to related to, to manifolds of core motifs one that Wolfgang Tiller and I developed, but let me just move on. So these are the examples and they have led to the, to the follow, these are constructions that led to the following examples. So to get started, you need, you need something, okay, I'll not write the main source for examples to this day, just leagrups, compact leagrups. Or leagrups with binary metrics, because they, if you take a binary metric, they belong to this class, okay? So therefore you can, you can, you can do all these constructions with them, okay? So you can take quotients. So all homogeneous spaces have metrics of non-negative curvature. The ones with positive curvature have been classified by Birard Bergieri and, and actually, so the examples constructed started with Marcel's work on the so-called normal homogeneous spaces. There are three of those, two in dimension seven, one dimension 13. And then Wallach constructed, classified the even dimensional examples in dimensions six, seven and 24. I have no idea if that 24 has to do with Brentelsenau, but I don't think so. But, and then Alaph and Wallach constructed a whole infinite family in dimension seven, okay? And that's it, all right? So, so, so if you look at homogeneous examples from this class, there are of course the crosses that were always there. And then they're the ones you have to work for to construct. And they exist in dimensions, I'll just write down the dimensions six, seven, 12, 13 and 24. 16 is one of the crosses, the Cayley plane kind of also unusual. But there is one example here, it's a flag manifold, one exam, another flag manifold, one exam, another flag manifold. This is a Berzei's example in dimension 13. And then there are infinite many here. Okay? Taking quotients, you can also take what's called bicochons, okay? So, so these are denoted. I really would like not to move on to another board. So, so a group eights can act on both left and right. So, we have a subgroup of d cross g acting on itself, maybe freely. And then these also have all have metric of non-negative curvature, of course. And there you get many more examples now with positive curvature. Okay? So, you get more examples in dimension seven and dimension 13. These are due to Esenberg, you have infinitely many. And here, due to Berzeichen, you have infinitely many examples. And that was where the situation stood up until recently. So, so these are, these are the constructions and examples that you have at your disposal. Of course, in non-negative curvature, you can do a lot more, right? You can take products, you can take quotients and this and that. So, that's why you get so many more examples. Now, so let me move on to see what, what you might do since, since if you're really not able to prove, let's say, Bruce is suggesting that you can find positive curve metrics on this two cross two arbitrary cross to the product metric or anything else, approve that hop the answer is no, you can't find such a metric or anything else. You've got to do something, right? And various approach, you could look at the pinching problem, pinch curvature, etc. It doesn't really need lead to new ideas, I would say. You can even look at the pinching of the size in terms of the diameter. Does, does also not lead to really new insights and new examples or anything. So, what you can do is, you can study all these examples and see if you get new ideas about manifold to positive curves or non-negative curves or whatnot. Or you could try to find more. Okay. One way to try to find more is to say, let's try to start over again and say, let's admit it. We can't do this in general yet. Let's just see what we can do in the presence of symmetries. So, like Bruce explained in dimension four, if you have a circle of isometries, you can actually prove what the manifold is. The manifold, if it has non-negative curvature, it's either S4, Cp2, S2 cross S2 connected some Cp2 plus minus Cp2. And you even know the action up to a very different morphism, complete classification. Okay. And so, you might, you might want, wonder what you can do in general. So, let me just write that down here. So, let me say, let's study the classes. m plus, n zero, and m zero minus in the presence of large or special isometric group actions. And it's not completely unreasonable to do it. Certainly, Broward's insight in dimension four, but all the known examples have a fair amount of symmetry, all of them, right? So, you could ask, is that how much does it take to classify? And it actually might give you new ideas to construct new examples as well. Okay. And this indeed has happened. So, let me just give you two examples of a notion of large. So, what could it mean that if a group G acts on m, that this group G is large? Well, one knows this obviously that the dimension of G is large relative to the dimension of m. This is called the degree of symmetry. Also studied by Wu Yixiang. I'd studied topologically by many people. You could think about the rank of G being large. So, you take a maximal torus. I just think of torus actions that play a role also in collapse manifold with bounded curvature. So, torus groups are certainly important. Another way of thinking of large is if the so-called core homogeneity, which is the dimension of the manifold, marginal of the group, so the orbit space, that this one is small. Okay. So, I will, oops, I will tell you about some results. Let's see. What should I do? I think I should bring back since we all, oh, I want that one down. Maybe now I'm really not so fast. Okay. All right. So, I will focus on just the rank condition. So, this one. So, suppose you have a torus that acts on a manifold m, isometrically. And here, in this situation, I'll assume m has positive curvature. And what you can show, for example, is in this case, is that the dimension of k cannot be too big. It can be in most roughly half of the dimension. And if it is, you can tell what it is, the manifold. It's actually like a cross, okay, up to diffeomorphism. Now, Wilking showed that you can do better. You can go down to about k is about one fourth of the dimension of m. And you can still say what the manifold is. It's rough again, like a cross, up to tangential homotopy equivalence. So, this equivalence is up to tangential homotopy equivalence. But there are also ways to address these questions due to Hopf in this setting. And Lee cannot, has a number of results related to this. Namely, answering basically, addressing the Hopf's questions. Forget there's two crosses, two. But think about the other ones for positive curvature. So, with k growing logarithmically as a function of n, as a logarithm function of n, there are many, there are many results. He has several results showing that then you cannot have a product, m not a product, and m, the Euler characteristic must be positive. So, what does this mean? So, of course, in this case, if here k grows linearly, it's like one, one fourth of the dimension, grows linearly, right? And then you can say a lot. You can say what the Betty numbers are and everything, right? So, in particular, this is true and they're also not products, right? That they are just like crosses, okay? But if you have much slower growth, in, so this is interesting at high dimensions, of course, in particular, then you could still conclude that you can't have manifolds with this amount of symmetry, torus symmetry, and, and m being a product of two manifolds, or examples like this. And also, you can conclude that the Euler characteristic is positive, okay? Now, here's the most spectacular work in this direction that just happened very recently, at least in my mind. So, so this is joint work due to Kenneth and Wilking. So, it simply says the following. If m is in this class, simply connect it, and t5 acts on m isometrically, then the Euler characteristic of m is at least two. So, this is, this is an amazing result in the sense that this is, doesn't matter what the dimension of m is, right? So, it has some symmetry, but it does, it tells, no matter how big dimensional manifold or positive courage you're looking at, it's one of those with positive Euler characteristics, and, and they can do more, apparently, than this, okay? This is proved in studying the fixed point sets of this and the topology of the fixed point sets. And so, this is, this is really something, okay? How am I, what am I supposed to stop? You can go a little further, but I think about, okay, so I will skip one part, but let me just say, let me just add then that, so in the other, in the other direction, and with the other notion of large, large points, they say the co-homogeneity. Let me, okay, let me just state the result in words. So, again, Wilking has a, has a, has a fundamental result that says if, if you fix the co-homogeneity to k, no matter what it is, could be 10 or whatever you want, then if the dimension is large enough, and it's not like one of these Gromov numbers, then the manifold is, is of the type of a cross, okay? If it has that co-homogeneity, okay? So, for example, look at the homogenous that are gone somewhere. How many, did, they dried up at dimensions 24, okay? And beyond that, they're only the crosses, okay? So, in co-homogeneity one, Zilla, and Wilking proved, gave sort of an, an exhaustive class of possibilities that is a classification in all dimensions, but seven, okay? In all dimensions, but seven, the result is the following. They are either across a normal homogenous space, or there are some infinite family of bzikin examples, and we know exactly what they are, okay? In dimensions seven, there are the, there are the two normal homogenous spaces. These are the only homogenous ones. And then there's an infinite family of the Eschenberg examples that also are co-homed to H1. And then there are two infinite families of candidates for positive curvature. That was what I was sort of driving at, right? And so, so in co-homogeneity one, they're like pk, qk candidates of new manifolds of positive curvature. I should say a means of that p1 is the seventh sphere, so this doesn't count. And q1 is one, the first alfamilic example, I call it 1, 1. The normal homogenous one actually. But p2 is known to be, now is in this class. This was proved by Derrick Hart, and independently by myself Verdiani and Ziller. By the way, Verdiani did the even-dimensional classified case before we did the odd-dimensional classification of these. Very recently, Derrick Hart has claimed that he can also do p3, q2, and q3. So these are new examples of manifolds of positive curvature, and so it sort of validates at least that this is not a bad approach to try to find new examples. And not even to speak of co-homogeneity two or whatnot. And I had meant to talk about co-homogeneity two, and the only reason was that there you can prove that if you are co-homogeneity two and in this last class, then you satisfy the bot conjecture, okay, then you're rasterly elliptic. I also would like to point out, and we'll finish by that. So if we go back to this list of examples, co-homogeneity zero means homogenous, we already saw that all the ones, they're all in M0, okay, I don't need to think about the almost non-negative curve. Co-homogeneity one, these are three distinct classes, okay, we have this very strong restriction to positive curvature. There are lots of examples that have non-zero curvature, but there are lots of examples also that do not have non-negative curvature, okay, but they all have almost non-negative curvature, right. In co-homogeneity two, most do not have almost non-negative curvature, period, but the ones that do, you can analyze, and in particular, they do, they are rasterly elliptic, okay, so let me stop here. Questions? From now, at least one question. What else is in the future? Yeah, well, I didn't have time. Well, what I, what I, I had a couple of things I want, so first of all, there's this question about why are only these crosses in high dimensions, what's going on? Maybe there's something to it, okay, or is it just because we are too ignorant and construct, all examples don't know how to construct examples other than using Lie groups, okay. Now, from, from that point of view, I think, so, so there's a lot of constructions now, we know much more now about a very interesting generalization of group axons, namely so-called singular Riemannian foliations, so these are locally everywhere equidistant, I mean, leaves, but they don't have to have the same dimension, but they are locally everywhere equidistant, these are called singular Riemannian foliations, and there's an abundance, no, Marco Rades has done a lot of really nice work on unit spheres, for example, not at all homogeneous, and so there's got to be a lot of examples of these in other, on other manifolds, right, and, and this recent work of Martin Kering, Chris Lanshanca and Gertze that showed that all exotic seven spheres, now we know that all exotic seven spheres have metrics of non-negative curvature, and this uses what Wolfgang and I did, but they took quotients, and one way of looking at the quotients that they looked at these constructions were they, that you can think of them as, as the co-homony to one picture with like a singular Riemannian foliation, where all the principal leaves are bicrotions instead of being homogeneous spaces, okay, so this is sort of like a hybrid between really general singular Riemannian foliations and homogeneous ones, right, but it shows that there are definitely interesting ways to think about these items in, in this context of singular Riemannian foliations, so, so I, I think that maybe one could construct examples that way, another one was there's a big old problem in, from the beginning of Alexandrov geometry that says that, that the boundary of an Alexandrov space should be an Alexandrov space with the same lower curvature bound, that's completely open, but recently Peterson and I and a student of mine, Adam Moreno, showed that it's true for leaf spaces of singular Riemannian foliations, so maybe another source is to look at all these general objects, take the boundary of such a thing, has non-negative curvature maybe, maybe positive curvature, in some cases they are manifold, maybe you can do deformations and so on and so forth, so those were some of the things I wanted to talk about, yeah. Well, if there are no further questions, thank you very much.
 Well first of all I'd like to express how honored I am to be part of this celebration  of Marcel Berger's life and work. So he had a tremendous impact on my own work, the  support he provided, support and interest, etc. meant a lot to me. And like Tom said,  we met in the early 70s, I met Berger in the early 70s and a whole group of his students.  I don't know if all of them were students but that was my sense. Since then I found out that  I looked up students on the math signet and also on the, that's right. And so it didn't look like  all the ones I knew. So I thought I knew many more and turns out that I think it's incomplete.  So, but independent of that I feel that, so he always had a large nice group of students  and from day one I felt like, oh these are so nice to be around and it's nice to be around Berger.  And their work and the whole attitude and support really meant a lot to me. In a certain sense I  feel like an orphan in Orhus because I grew up with topologists at the time. They were very,  very algebraic topologists. So I broke from that and I wanted to do geometry. And so Marcel Berger  was really my idol and then clinging back to the big main figures in Europe and the world  actually at that point. So I thought I would talk about, I took also the title of the conference  literary. So I wanted to give like a survey, talk about past, present and future. On one of the  topics where Marcel made a tremendous impact, namely manifold to positive curvature. So let me  tell you immediately. So I want to focus on, so I'll write this as M plus. So  manifolds in M plus are simply manifolds with sexual curvature positive. But here comes the  beyond part. So the beyond part is I cannot help also to talk about manifolds of non-negative  sexual curvature. And I may even go a little bit further because they're very natural classes  related to one another. Namely those that are called manifold of almost non-negative curvature.  For now, I'll just write it like this and I'll explain what this means in a little bit.  So this of course is really where Berger has tremendous impact.  And much of my own work has been concerned with this area. But I think I'll focus as much as I  can on generalities in this talk. So I will focus general results as much as I can.  And then things stop quickly actually. But still,  now, so let me just remind you geometrically what this means. So one way to think about this  is simply the following. If you take two vectors at a point P and you look at the corresponding  geodesics, as long as they are minimal, the curvature positive means that the distance  between the endpoints is smaller than the distance between the endpoints in the tangent space.  And greater equal just means that they are less than or equal to. So this is the Hinsch version  of Topologov's theorem. There are many equivalent formulations of that theorem. And all of this  started with these comparisons theorems, comparisons of Jacobi fields. The Rauch-1 theorem, it starts  out comparison of Jacobi fields that start out by being zero at a point. And then there's Rauch-2.  I don't know why it's called Rauch-2 because it's due to Berger. So this is about Jacobi fields that  start off by having some value, but covariant derivative zero at the point. And those two  are really basic results that led to the proof of Topologov's theorem. So now,  okay, so I think I will use this board. No? The reason I'm doing this is I thought I had figured  it all out to avoid these shadows, but it didn't quite work. At any rate, let me start with the  most general results in this, if I take the area as a whole, okay? So there are two, I don't think  anyone gets offended. There are two milestone results of general type. One of them is due to  Chigur and Gromal. It's a Sol theorem. Originally, Wolfgang Meier did some previous work with Gromal  also. So it simply says the following. If m is a manifold with non-negative curvature, I assume  they're complete all of them because otherwise we know for Gromal there's no theorems, right?  So m is always complete but non-compact.  This implies that m is actually the total space of a compact manifold that sits in isometrically.  So m, so there exists a compact sub-manifold and this is totally convex,  meaning that any geodesic starting and ending at s is in s. So in particular, it's really  metrically embedded in here, totally geodesic, such that m is diffeomorphic to the normal bundle.  So the total space of the normal bundle and this equivalence is up to diffeomorphism.  All right, so this is one theorem. This is, you can think of it as, I mean, I like to think of it  as a structure theorem or reduction theorem if you like. It's a structure theorem because it tells  you what such manifolds look like, what's the structure of these manifolds. So they're all  vector bundles over something compact from the same category. So it follows here that of course s  is also in the same class and I should add that this is what's called the sole, of course.  If m happens to be in this class, if it happens to have positive curvature, then the sole is a point.  And in other words, then m is just diffeomorphic to rn, which means that I won't talk about it anymore.  So we have a complete classification of positively curved manifolds that are non-compact  in the sense of the diffeomorphism type. So in that sense, it's also a reduction theorem  because now we're reduced to think about compact manifolds. So for the rest of the talk, I'll talk  about compact manifolds in these classes. But before I move on to this, there's an obvious  set of problems related to this, namely the converse problem. So you can imagine, given  so let's see how this is working. Maybe it's not working at all.  Oh, no, it's not working. Only if I put it all the way up.  No, this is not going to work. I thought I had these brilliant ideas.  You see, they teach us things that I never learned in any other place but Notre Dame.  If there are two blackboards, they teach you from day one, I think, that you should always start  with the one that's behind because of the shadow issue. I can't even see it now,  so maybe I think I'll just give up and you will have to live with that shadow.  Okay, so I'll pull it down again because this becomes much more comfortable for me,  but now I started with the back button. God. All right, I'll just say it in words, okay?  I'll just move on to something else. So what's the converse? The converse problem is you give  yourself a compact manifold of non-negative curvature and a vector bundle over it. The question is,  can you put a complete metric of non-negative curvature on the total space? Okay, now this is  known to be false in general. There are lots of examples, the first were due to Waldschap and Oseiden  and many more since then due to Bellegrad and Kapovitz. All of those examples are examples with  infinite fundamental group. No counter example is known for finite, in particular, simply connected  manifolds. Now, so and the original question by Chig and Grimal was if what about bundles over spheres?  Even that is not known, only in dimensions up to the base being five-dimensional.  However, Rieger's a long time ago in his thesis proved that over spheres,  stably it's true. Okay, so you can add a large enough trivial bundle and then it's true. Okay,  you can put a metric of non-negative curvature on it and recently it's been a lot of work by  David Gonzalez about all other kinds of manifolds where you stable it's true. In particular, I  think by now it's known to take any known positive recurve manifold then it's true all vector bundles  over those if you add a trivial bundle. Okay, so that's a reasonable general question. Okay.  All right, so this was the converse of that. So now let me pull this board down again  and write down the other sort of general result.  Oh, I can't do that now, right? Because I'm not tall enough. Yeah.  Yeah. This is taking more time than I planned. When I came I saw all these boards I said,  oh, I can finish this talk in no time. Okay. Yeah, I really messed up with this here.  All right, so what's the other one? You probably guessed it. So this is the Benin-Uber theorem.  It's a theorem due to Gromov and it simply says so like I said from now on we  focus on compact manifolds. So one version of it is the following. If mn, so it exists  a constant depending on the dimension n such that if m is in this class of non-negative  curvature then the sum or the dimension of the homology of m, any field of coefficients  is bounded by this constant n. Okay. Now there's a more general version of this theorem  that doesn't care about what the lower curvature bound is and I should also add that the same  thing is true for the fundamental group, pi1 of m is generated by less than some constant also.  This one is a very simple beautiful consequence of Tobinogov's theorem. This one is really hard.  It's a deep theorem and it's sort of the starting point is actually a rough version of this theorem  that says that the topology so but it was proved only after the sole theorem but if you look at a  complete manifold of non-negative curvature then it's a compact type. So that was Gromov's  observation meaning that if you go far enough away there are no critical points for the distance  function to the sole. So that means that it has the type of a compact manifold with boundary.  That really is one of the key starting points in his proof of this thing for compact manifolds.  Okay. But I won't say anything more about it. So now the more general result is for any curvature  bound and then a bound also on the diameter. Of course in non-negative curvature you don't see  that because you can always scale it. You can even go as far as you could say whatever we're  going to discuss from now on we scale the diameter to be some number. Okay. Maybe pi. We like pi for  the unit sphere maybe or else and then it's just a question about what is the lower curvature  with the minimum of the lower bound for the curvature. Right. That's like a one function  of this class of manifolds and this theorem says that if that function is greater than equal to zero  then this is true but there's a value for all of them. Okay. There's a value that depends on this  number. Okay. However that value is really large and so in particular if we go back to this  general situation it does not distinguish these classes. Okay. You can you could say if you look  at this thing this it's kind of the same number. There's no at this point. Okay.  So one of the huge problems I would say that is is that we don't even know that these classes are  different while we do at the level of fundamental groups. Okay. So at the level of fundamental  groups these are really distinct classes. Here for example we have Born-Emeyer's theorem.  The classical theorem. So now this is positive curvature. That's a general theorem of positive  curvature. So m in m plus implies pi one of m is finite. This is the classical Born-Emeyer's  theorem. It's true for retrocurvature even. And of course this is not true for the other classes.  Okay. But so but actually you know what the structure is. You know a lot about the structure  manifolds in these other classes. So for example in the class m zero you know that any compact  manifold has a finite cover which is at least diffeomorphic to a simply connected one with  non-negative terms across the torus. Okay. And what is this last class? Maybe I now should tell  you what this means. So what does it mean that m is in this class? Well it means a couple of ways  to say it. One is to say that there exists a family of metrics. Can you read this by the way up there?  Well that's impressive because my handwriting is bad too. But I'll try to say it out loud. So there's a  family of metrics on m so that the section curvature of m relative to this metric is greater than or  equal to minus one. And the diameter of m with this metric approaches zero. That's one way of saying  it. So this means collapse with lower curvature bound. Okay. This is equivalent by rescaling.  You can see that there exists g i. Here's another way of saying it. So the sexual curvature of m i  m g i is greater than or equal to let's say minus one divided by i where i goes to infinity. So this  approach is zero getting closer and closer to zero. And let's say the diameter of m g i is a fixed  number maybe pi. Okay. So this is why we call them almost non-negative and correct of course.  And many folks in this class have not been studied that much but I think they really are important  for understanding the whole picture and for understanding in general manifolds with lower  curvature bound. Okay. Just like in the case of manifolds with bounded curvature,  where almost flat manifolds play a central role in the work of Chica Foucaille and Gromov. Okay.  So that's expected but hasn't been really developed.  All right. By the way in this class any manifold in this class is also finally covered  by a compact manifold that fibers over a nil manifold with fiber being simply connected.  Okay. But in general the class of fundamental group is larger. But if we move on to assume  that m is simply connected then there's not a single example known that of even going all the way  to almost non-negative curvature that couldn't possibly have positive curvature. No obstructions  are known. Okay. Let me add so this is so what are the other general theorems about positive  curvature? They also go back a long time. They go back to synths or sing. So it's also about just  about the fundamental group. If m is even dimensional this implies that pi1 is either trivial or z mod  2. This is when m is orientable and this is of course when it's not. Okay. And if it is odd  dimensional then m is orientable. And these are results going back to synth in the 30s. Okay.  And all of them are sort of like baby Morse theory in the sense.  You look at the minimum general length of general minimal shortest closed deodesic or whatever in  the homotopic class and you show that it has to be trivial. Okay. So this is just a simple variation  formula. So in this example here you take a suppose m is not simply connected. You take a  shortest closed deodesic that's non-trivial. You find from the assumptions a parallel field  and you just look at the variation in that direction and then you find shorter curves.  Okay. There are several theorems like this and I actually all put them in the same block. I kind  them synth type theorems. Okay. There's a theorem due to Frankel that says in positive curvature if  you have two totally deodesic sub manifolds they intersect if their dimensions are too big.  It's exactly the same proof kind of going back to Weinstein if you have an isometry and such a  manifold you find fixed points in even the mentions if it preserves orientation. And that by the way  is very similar to Berger's theorem about zeros of killing fields. Okay. So he proved that any  killing field on an even dimensional manifold has a zero and in order mentions it's true that if you  have two killing fields they're dependent at one point. Okay. So these are the general results  that I know of for manifolds of positive curvature without any further assumptions. Okay. So that's  end of story in that sense. Okay. There's much more work that has been done but all of it with  additional assumptions. So for example what what Marcel did of course was to prove the quarter  pinching theorem and he proved other things very interesting things positive curvature.  But these are not like completely general results. Okay. So I'm trying to stick with  completely general results at this point and see what what else one can do.  All right. So now I want to move on to some basic conjectures.  And this is really not working well.  Okay.  Okay.  Can you see it? Okay. I know the theorem so I don't need to see it. Okay. I can't see it.  All right. So what are some main conjectures?  Maybe I'll give up and just put this one here and then then do it reverse afterwards.  Okay. So we can't avoid Hopf and there was mentioned by Bruce earlier today.  So the Hopf has two basic conjectures attributed to Heinz-Hopf. One of them is that the Euler  characteristic of M is greater than or equal to zero when M is non-negatively curved and actually  it should be positive when M has positive sectional curvature. And the other one, the one that Bruce  mentioned is the question whether S2 cross S2 is in or not. I guess if it's a conjecture,  the conjecture was if it's not, but maybe Hopf asked just asked the question. So,  okay. Now this has two very natural more general related questions I would say. So one is the  following. If you take two manifolds N and V and take their product, there is not a single  such example known to have positive curvature. I'm not saying conduct time anything. I'm just saying  there's not such an example known yet. So of course, here's what we're talking about as 2 cross S2.  Well, I mean this is not, if I take them to be simply connected, okay, take them to be simply  connected. So only Rp2 cross Rp2 doesn't have positive curvature. So the zoom N and V is simply  connected. And the other direction is how about you could think about symmetric spaces  and I'll use, I guess this goes back to best this notation maybe, I think so, right. Compact rank  one symmetric space, okay. So S2 cross S2 has ranked two, okay. And you could ask this question  for any symmetric space a rank at least two. Is there a not right. Plus I'm saying it shouldn't  have positive curvature. Oh, yes. Oh, then I can't attribute this. I just wanted to do that.  Symmetric space, darn. Symmetric space, thank you. Of rank greater than or equal to two. Yes,  thank you. Of course, that's what I meant. Yes, the other ones do. All right, so these are  basic questions and here's a basic conjecture due to bot. That says that the, if we look at the,  if M is also in this class, then the sequence of bedding numbers of the loop space of M  grows at most  polynomially. Okay. There's, there's a, this is, I'm not sure this is the formulation bot had  because it's not in any, it's not written down, but bot wrote a paper with the Berger  way where they looked at pinched manifold. So in this class with two bounds of the curvature  and they provided estimates for the bedding numbers, but they're way from this way off from  there and the exponential growth. They are estimates in terms of the pinching, but they  have nothing to do with this conclusion that you would like. Okay. Now, if you focus, so you could  also, another question is, are you thinking of any field like in Gromov's theorem or what not? Okay.  But if you, if you, so here we could say F and this could be any field, maybe. That's a reasonable  conjecture, but it becomes really obviously interesting in particular. There's a lot of  consequences. If you take F to be the rational numbers, because then this condition is actually  equivalent to the following that the dimension of the homotropic groups, the rational homotropic groups  is finite. Okay. So this follows via the minimal model theory of Solomon and so on and so forth.  This is old also, right? This is also way past. Okay. Now, this by itself, I'll just write down,  it has lots of consequences that have been proved, topologically. One of them is that the Euler  characteristic is then greater than or equal to zero. This follows from this. Okay. So, so the butt  conjecture would imply one part of the Huff conjecture at least. The other part is that it's, and so,  so which suggests what to try to do if you want to prove the Huff conjecture, if you believe  butts. So, because being positive for the Euler characteristic is then equivalent to all odd  rational better numbers to be zero. Okay. So in a sense, you might as well just prove that to prove  the Huff conjecture. All right. So, all right. Another consequence is related to Gromov's theorem,  and it is that the sum of better numbers, rational better numbers here, is at most two to the end  where n is the dimension. See, that's a good bound, right? And in general, to distinguish these  classes, one might want to prove just bounds that would allow you to distinguish them. That's what  maybe one possibility, but that's, that's really all I wanted to mention here. Okay. Now,  so, given this bleak situation about general results, I'd like to provide some other bleak  news about how many examples do we know of this? Okay. So examples,  and constructions.  So what is, what are, how do we construct examples? Well, the, so let me talk about some constructions  first. So if you start with a manifold m in m zero, and we look at, at, I'll call these projections.  So you look at, at, at a, like a Croson map, m to n pi, where this is a Riemannian submersion,  then this implies that this guy is also in this class, and you might be clever or lucky or whatever,  and show that sometimes it is in this class. Okay. That's a very important construction. Now, if you  look at, if you look at this class, Tom and I went out together last time, I'm losing my wise two now.  If you look at this class, then you can take products, right? Products in this class work,  of course. And then there's the bundles, and I would say in this class, what do I mean by bundles?  It's the, that's the converse of this one in a certain sense. This is a very hard direction to go in.  So basically, you start with something, you start with, with something, a base in a class,  and you have a bundle over it, maybe a principal bundle, maybe a principal G bundle. I'll just stick  to that right now. Then it follows that when B is in this class, that P is also in this class.  So this is a way to construct many examples of almost non-negative curve manifolds that doesn't  work in either of these categories in general, right? That's why this inverse, the converse of  the sole theorem is very hard, right? So to go up to bundles is very difficult, right, to construct  metrics. So these are some of the constructions. There's one other construction, and this is  called the special gluings. And Jeff was the first to make such interesting constructions,  namely, for example, and now, now I can use my word cross, okay? So if you take m and n,  okay, compact rank one symmetric spaces, then  Cheeg approved that their connected sum has a metric of non-negative curvature, okay?  Basically using the, if you cut out a point from one of these space, you have a bundle  and you change it so that it becomes product near the, near the boundary so that you can glue them  together isometrically with a, by means of unit sphere. And there are other constructions about  gluings that have to related to, to manifolds of core motifs one that Wolfgang Tiller and I developed,  but let me just move on. So these are the examples and they have led to the, to the  follow, these are constructions that led to the following examples. So to get started, you need,  you need something, okay, I'll not write the main source for examples to this day,  just leagrups, compact leagrups.  Or leagrups with binary metrics, because they, if you take a binary metric,  they belong to this class, okay? So therefore you can, you can, you can do all these constructions  with them, okay? So you can take quotients. So all homogeneous spaces have metrics of  non-negative curvature. The ones with positive curvature have been classified by Birard Bergieri  and, and actually, so the examples constructed started with Marcel's work on the so-called  normal homogeneous spaces. There are three of those, two in dimension seven, one dimension 13.  And then Wallach constructed, classified the even dimensional examples in dimensions six,  seven and 24. I have no idea if that 24 has to do with Brentelsenau, but I don't think so. But,  and then Alaph and Wallach constructed a whole infinite family in dimension seven, okay? And  that's it, all right? So, so, so if you look at homogeneous examples from this class,  there are of course the crosses that were always there. And then they're the ones you  have to work for to construct. And they exist in dimensions, I'll just write down the dimensions  six, seven, 12, 13 and 24. 16 is one of the crosses, the Cayley plane kind of also unusual. But there  is one example here, it's a flag manifold, one exam, another flag manifold, one exam, another flag  manifold. This is a Berzei's example in dimension 13. And then there are infinite many here. Okay?  Taking quotients, you can also take what's called bicochons, okay? So, so these are denoted.  I really would like not to move on to another board. So,  so a group eights can act on both left and right. So, we have a subgroup of d cross g acting on  itself, maybe freely. And then these also have all have metric of non-negative curvature, of  course. And there you get many more examples now with positive curvature. Okay? So, you get more  examples in dimension seven and dimension 13. These are due to Esenberg, you have infinitely many.  And here, due to Berzeichen, you have infinitely many examples. And that was where the situation  stood up until recently. So, so these are, these are the constructions and examples that you have  at your disposal. Of course, in non-negative curvature, you can do a lot more, right? You can  take products, you can take quotients and this and that. So, that's why you get so many more examples.  Now, so let me move on to see what, what you might do since, since if you're really not able to prove,  let's say, Bruce is suggesting that you can find positive curve metrics on this two cross two  arbitrary cross to the product metric or anything else, approve that hop the answer is no, you can't  find such a metric or anything else. You've got to do something, right? And various approach, you  could look at the pinching problem, pinch curvature, etc. It doesn't really need lead to new ideas,  I would say. You can even look at the pinching of the size in terms of the diameter. Does,  does also not lead to really new insights and new examples or anything. So, what you can do is,  you can study all these examples and see if you get new ideas about manifold to positive curves  or non-negative curves or whatnot. Or you could try to find more. Okay. One way to try to find more  is to say, let's try to start over again and say, let's admit it. We can't do this in general yet.  Let's just see what we can do in the presence of symmetries. So, like Bruce explained in dimension  four, if you have a circle of isometries, you can actually prove what the manifold is. The manifold,  if it has non-negative curvature, it's either S4, Cp2, S2 cross S2 connected some Cp2 plus  minus Cp2. And you even know the action up to a very different morphism, complete classification.  Okay. And so, you might, you might want, wonder what you can do in general. So, let me just write  that down here. So, let me say, let's study the classes. m plus, n zero, and m zero minus in the  presence of large or special isometric group actions. And it's not completely unreasonable to do it.  Certainly, Broward's insight in dimension four, but all the known examples have a fair amount of  symmetry, all of them, right? So, you could ask, is that how much does it take to classify? And it  actually might give you new ideas to construct new examples as well. Okay. And this indeed has happened.  So, let me just give you two examples of a notion of large. So, what could it mean that if a group  G acts on m, that this group G is large? Well, one knows this obviously that the dimension of G is  large relative to the dimension of m. This is called the degree of symmetry. Also studied by Wu Yixiang.  I'd studied topologically by many people. You could think about the rank of G being large. So,  you take a maximal torus. I just think of torus actions that play a role also in collapse manifold  with bounded curvature. So, torus groups are certainly important. Another way of thinking of large  is if the so-called core homogeneity, which is the dimension of the manifold,  marginal of the group, so the orbit space, that this one is small. Okay. So, I will,  oops, I will tell you about some results.  Let's see. What should I do? I think I should bring back since we all, oh, I want that one down.  Maybe now I'm really not so fast.  Okay.  All right. So, I will focus on just the rank condition. So, this one.  So, suppose you have a torus that acts on a manifold m,  isometrically. And here, in this situation, I'll assume m has positive curvature.  And what you can show, for example, is in this case, is that the dimension of k cannot be too big.  It can be in most roughly half of the dimension. And if it is, you can tell what it is, the  manifold. It's actually like a cross, okay, up to diffeomorphism. Now,  Wilking showed that you can do better. You can go down to about k is about one fourth of the  dimension of m. And you can still say what the manifold is. It's rough again, like a cross,  up to tangential homotopy equivalence. So, this equivalence is up to tangential homotopy equivalence.  But there are also ways to address these questions due to Hopf in this setting.  And Lee cannot, has a number of results related to this. Namely, answering basically, addressing  the Hopf's questions. Forget there's two crosses, two. But think about the other ones for positive  curvature. So, with k growing logarithmically as a function of n, as a logarithm function of n,  there are many, there are many results. He has several results showing that then you cannot  have a product, m not a product, and m, the Euler characteristic must be positive. So,  what does this mean? So, of course, in this case, if here k grows linearly, it's like one,  one fourth of the dimension, grows linearly, right? And then you can say a lot. You can say  what the Betty numbers are and everything, right? So, in particular, this is true and they're also  not products, right? That they are just like crosses, okay? But if you have much slower growth,  in, so this is interesting at high dimensions, of course, in particular, then you could still  conclude that you can't have manifolds with this amount of symmetry, torus symmetry, and, and m being  a product of two manifolds, or examples like this. And also, you can conclude that the Euler characteristic  is positive, okay? Now, here's the most spectacular work in this direction that just happened very  recently, at least in my mind. So, so this is joint work due to Kenneth and Wilking.  So, it simply says the following. If m is in this class, simply connect it,  and t5 acts on m isometrically,  then the Euler characteristic of m is at least two. So, this is, this is an amazing result in the  sense that this is, doesn't matter what the dimension of m is, right? So, it has some symmetry,  but it does, it tells, no matter how big dimensional manifold or positive courage you're looking at,  it's one of those with positive Euler characteristics, and, and they can do more, apparently, than this,  okay? This is proved in studying the fixed point sets of this and the topology of the fixed point  sets. And so, this is, this is really something, okay? How am I, what am I supposed to stop?  You can go a little further, but I think about, okay, so I will skip one part, but let me just say,  let me just add then that, so in the other, in the other direction, and with the other notion of large,  large points, they say the co-homogeneity.  Let me, okay, let me just state the result in words. So, again, Wilking has a, has a, has a  fundamental result that says if, if you fix the co-homogeneity to k, no matter what it is, could be  10 or whatever you want, then if the dimension is large enough, and it's not like one of these  Gromov numbers, then the manifold is, is of the type of a cross, okay? If it has that co-homogeneity,  okay? So, for example, look at the homogenous that are gone somewhere. How many, did, they dried  up at dimensions 24, okay? And beyond that, they're only the crosses, okay? So, in co-homogeneity one,  Zilla, and Wilking proved, gave sort of an, an exhaustive class of possibilities that is a  classification in all dimensions, but seven, okay? In all dimensions, but seven, the result is the  following. They are either across a normal homogenous space, or there are some infinite  family of bzikin examples, and we know exactly what they are, okay? In dimensions seven,  there are the, there are the two normal homogenous spaces. These are the only homogenous ones.  And then there's an infinite family of the Eschenberg examples that also are co-homed to H1.  And then there are two infinite families of candidates for positive curvature. That was what I  was sort of driving at, right? And so, so in co-homogeneity one, they're like pk, qk candidates  of new manifolds of positive curvature. I should say a means of that p1 is the seventh sphere,  so this doesn't count. And q1 is one, the first alfamilic example, I call it 1, 1. The normal  homogenous one actually. But p2 is known to be, now is in this class. This was proved by Derrick  Hart, and independently by myself Verdiani and Ziller. By the way, Verdiani did the even-dimensional  classified case before we did the odd-dimensional classification of these. Very recently, Derrick  Hart has claimed that he can also do p3, q2, and q3. So these are new examples of manifolds  of positive curvature, and so it sort of validates at least that this is not a bad approach to try  to find new examples. And not even to speak of co-homogeneity two or whatnot. And I had meant  to talk about co-homogeneity two, and the only reason was that there you can prove that if you  are co-homogeneity two and in this last class, then you satisfy the bot conjecture, okay,  then you're rasterly elliptic. I also would like to point out, and we'll finish by that. So if we  go back to this list of examples, co-homogeneity zero means homogenous, we already saw that  all the ones, they're all in M0, okay, I don't need to think about the almost non-negative curve.  Co-homogeneity one, these are three distinct classes, okay, we have this very strong restriction  to positive curvature. There are lots of examples that have non-zero curvature,  but there are lots of examples also that do not have non-negative curvature, okay, but they all  have almost non-negative curvature, right. In co-homogeneity two, most do not have almost  non-negative curvature, period, but the ones that do, you can analyze, and in particular,  they do, they are rasterly elliptic, okay, so let me stop here.  Questions?  From now, at least one question.  What else is in the future?  Yeah, well, I didn't have time. Well, what I, what I, I had a couple of things I want, so first  of all, there's this question about why are only these crosses in high dimensions, what's going on?  Maybe there's something to it, okay, or is it just because we are too ignorant and construct,  all examples don't know how to construct examples other than using Lie groups, okay. Now, from,  from that point of view, I think, so, so there's a lot of constructions now, we know much more now  about a very interesting generalization of group axons, namely so-called singular Riemannian  foliations, so these are locally everywhere equidistant, I mean, leaves, but they don't have  to have the same dimension, but they are locally everywhere equidistant, these are called singular  Riemannian foliations, and there's an abundance, no, Marco Rades has done a lot of really nice work  on unit spheres, for example, not at all homogeneous, and so there's got to be a lot of examples of these  in other, on other manifolds, right, and, and this recent work of Martin Kering,  Chris Lanshanca and Gertze that showed that all exotic seven spheres, now we know that all  exotic seven spheres have metrics of non-negative curvature, and this uses what Wolfgang and I did,  but they took quotients, and one way of looking at the quotients that they looked at these constructions  were they, that you can think of them as, as the co-homony to one picture with like a singular  Riemannian foliation, where all the principal leaves are bicrotions instead of being homogeneous  spaces, okay, so this is sort of like a hybrid between really general singular Riemannian foliations  and homogeneous ones, right, but it shows that there are definitely interesting ways to think about  these items in, in this context of singular Riemannian foliations, so, so I, I think that  maybe one could construct examples that way, another one was there's a big old problem in,  from the beginning of Alexandrov geometry that says that, that the boundary of an Alexandrov space  should be an Alexandrov space with the same lower curvature bound, that's completely open,  but recently Peterson and I and a student of mine, Adam Moreno, showed that it's true for  leaf spaces of singular Riemannian foliations, so maybe another source is to look at all these  general objects, take the boundary of such a thing, has non-negative curvature maybe, maybe  positive curvature, in some cases they are manifold, maybe you can do deformations and so on and so  forth, so those were some of the things I wanted to talk about, yeah.  Well, if there are no further questions, thank you very much.
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Well first of all I'd like to express how honored I am to be part of this celebration of Marcel Berger's life and work. So he had a tremendous impact on my own work, the support he provided, support and interest, etc. meant a lot to me. And like Tom said, we met in the early 70s, I met Berger in the early 70s and a whole group of his students. I don't know if all of them were students but that was my sense. Since then I found out that I looked up students on the math signet and also on the, that's right. And so it didn't look like all the ones I knew. So I thought I knew many more and turns out that I think it's incomplete. So, but independent of that I feel that, so he always had a large nice group of students and from day one I felt like, oh these are so nice to be around and it's nice to be around Berger. And their work and the whole attitude and support really meant a lot to me. In a certain sense I feel like an orphan in Orhus because I grew up with topologists at the time. They were very, very algebraic topologists. So I broke from that and I wanted to do geometry. And so Marcel Berger was really my idol and then clinging back to the big main figures in Europe and the world actually at that point. So I thought I would talk about, I took also the title of the conference literary. So I wanted to give like a survey, talk about past, present and future. On one of the topics where Marcel made a tremendous impact, namely manifold to positive curvature. So let me tell you immediately. So I want to focus on, so I'll write this as M plus. So manifolds in M plus are simply manifolds with sexual curvature positive. But here comes the beyond part. So the beyond part is I cannot help also to talk about manifolds of non-negative sexual curvature. And I may even go a little bit further because they're very natural classes related to one another. Namely those that are called manifold of almost non-negative curvature. For now, I'll just write it like this and I'll explain what this means in a little bit. So this of course is really where Berger has tremendous impact. And much of my own work has been concerned with this area. But I think I'll focus as much as I can on generalities in this talk. So I will focus general results as much as I can. And then things stop quickly actually. But still, now, so let me just remind you geometrically what this means. So one way to think about this is simply the following. If you take two vectors at a point P and you look at the corresponding geodesics, as long as they are minimal, the curvature positive means that the distance between the endpoints is smaller than the distance between the endpoints in the tangent space. And greater equal just means that they are less than or equal to. So this is the Hinsch version of Topologov's theorem. There are many equivalent formulations of that theorem. And all of this started with these comparisons theorems, comparisons of Jacobi fields. The Rauch-1 theorem, it starts out comparison of Jacobi fields that start out by being zero at a point. And then there's Rauch-2. I don't know why it's called Rauch-2 because it's due to Berger. So this is about Jacobi fields that start off by having some value, but covariant derivative zero at the point. And those two are really basic results that led to the proof of Topologov's theorem. So now, okay, so I think I will use this board. No? The reason I'm doing this is I thought I had figured it all out to avoid these shadows, but it didn't quite work. At any rate, let me start with the most general results in this, if I take the area as a whole, okay? So there are two, I don't think anyone gets offended. There are two milestone results of general type. One of them is due to Chigur and Gromal. It's a Sol theorem. Originally, Wolfgang Meier did some previous work with Gromal also. So it simply says the following. If m is a manifold with non-negative curvature, I assume they're complete all of them because otherwise we know for Gromal there's no theorems, right? So m is always complete but non-compact. This implies that m is actually the total space of a compact manifold that sits in isometrically. So m, so there exists a compact sub-manifold and this is totally convex, meaning that any geodesic starting and ending at s is in s. So in particular, it's really metrically embedded in here, totally geodesic, such that m is diffeomorphic to the normal bundle. So the total space of the normal bundle and this equivalence is up to diffeomorphism. All right, so this is one theorem. This is, you can think of it as, I mean, I like to think of it as a structure theorem or reduction theorem if you like. It's a structure theorem because it tells you what such manifolds look like, what's the structure of these manifolds. So they're all vector bundles over something compact from the same category. So it follows here that of course s is also in the same class and I should add that this is what's called the sole, of course. If m happens to be in this class, if it happens to have positive curvature, then the sole is a point. And in other words, then m is just diffeomorphic to rn, which means that I won't talk about it anymore. So we have a complete classification of positively curved manifolds that are non-compact in the sense of the diffeomorphism type. So in that sense, it's also a reduction theorem because now we're reduced to think about compact manifolds. So for the rest of the talk, I'll talk about compact manifolds in these classes. But before I move on to this, there's an obvious set of problems related to this, namely the converse problem. So you can imagine, given so let's see how this is working. Maybe it's not working at all. Oh, no, it's not working. Only if I put it all the way up. No, this is not going to work. I thought I had these brilliant ideas. You see, they teach us things that I never learned in any other place but Notre Dame. If there are two blackboards, they teach you from day one, I think, that you should always start with the one that's behind because of the shadow issue. I can't even see it now, so maybe I think I'll just give up and you will have to live with that shadow. Okay, so I'll pull it down again because this becomes much more comfortable for me, but now I started with the back button. God. All right, I'll just say it in words, okay? I'll just move on to something else. So what's the converse? The converse problem is you give yourself a compact manifold of non-negative curvature and a vector bundle over it. The question is, can you put a complete metric of non-negative curvature on the total space? Okay, now this is known to be false in general. There are lots of examples, the first were due to Waldschap and Oseiden and many more since then due to Bellegrad and Kapovitz. All of those examples are examples with infinite fundamental group. No counter example is known for finite, in particular, simply connected manifolds. Now, so and the original question by Chig and Grimal was if what about bundles over spheres? Even that is not known, only in dimensions up to the base being five-dimensional. However, Rieger's a long time ago in his thesis proved that over spheres, stably it's true. Okay, so you can add a large enough trivial bundle and then it's true. Okay, you can put a metric of non-negative curvature on it and recently it's been a lot of work by David Gonzalez about all other kinds of manifolds where you stable it's true. In particular, I think by now it's known to take any known positive recurve manifold then it's true all vector bundles over those if you add a trivial bundle. Okay, so that's a reasonable general question. Okay. All right, so this was the converse of that. So now let me pull this board down again and write down the other sort of general result. Oh, I can't do that now, right? Because I'm not tall enough. Yeah. Yeah. This is taking more time than I planned. When I came I saw all these boards I said, oh, I can finish this talk in no time. Okay. Yeah, I really messed up with this here. All right, so what's the other one? You probably guessed it. So this is the Benin-Uber theorem. It's a theorem due to Gromov and it simply says so like I said from now on we focus on compact manifolds. So one version of it is the following. If mn, so it exists a constant depending on the dimension n such that if m is in this class of non-negative curvature then the sum or the dimension of the homology of m, any field of coefficients is bounded by this constant n. Okay. Now there's a more general version of this theorem that doesn't care about what the lower curvature bound is and I should also add that the same thing is true for the fundamental group, pi1 of m is generated by less than some constant also. This one is a very simple beautiful consequence of Tobinogov's theorem. This one is really hard. It's a deep theorem and it's sort of the starting point is actually a rough version of this theorem that says that the topology so but it was proved only after the sole theorem but if you look at a complete manifold of non-negative curvature then it's a compact type. So that was Gromov's observation meaning that if you go far enough away there are no critical points for the distance function to the sole. So that means that it has the type of a compact manifold with boundary. That really is one of the key starting points in his proof of this thing for compact manifolds. Okay. But I won't say anything more about it. So now the more general result is for any curvature bound and then a bound also on the diameter. Of course in non-negative curvature you don't see that because you can always scale it. You can even go as far as you could say whatever we're going to discuss from now on we scale the diameter to be some number. Okay. Maybe pi. We like pi for the unit sphere maybe or else and then it's just a question about what is the lower curvature with the minimum of the lower bound for the curvature. Right. That's like a one function of this class of manifolds and this theorem says that if that function is greater than equal to zero then this is true but there's a value for all of them. Okay. There's a value that depends on this number. Okay. However that value is really large and so in particular if we go back to this general situation it does not distinguish these classes. Okay. You can you could say if you look at this thing this it's kind of the same number. There's no at this point. Okay. So one of the huge problems I would say that is is that we don't even know that these classes are different while we do at the level of fundamental groups. Okay. So at the level of fundamental groups these are really distinct classes. Here for example we have Born-Emeyer's theorem. The classical theorem. So now this is positive curvature. That's a general theorem of positive curvature. So m in m plus implies pi one of m is finite. This is the classical Born-Emeyer's theorem. It's true for retrocurvature even. And of course this is not true for the other classes. Okay. But so but actually you know what the structure is. You know a lot about the structure manifolds in these other classes. So for example in the class m zero you know that any compact manifold has a finite cover which is at least diffeomorphic to a simply connected one with non-negative terms across the torus. Okay. And what is this last class? Maybe I now should tell you what this means. So what does it mean that m is in this class? Well it means a couple of ways to say it. One is to say that there exists a family of metrics. Can you read this by the way up there? Well that's impressive because my handwriting is bad too. But I'll try to say it out loud. So there's a family of metrics on m so that the section curvature of m relative to this metric is greater than or equal to minus one. And the diameter of m with this metric approaches zero. That's one way of saying it. So this means collapse with lower curvature bound. Okay. This is equivalent by rescaling. You can see that there exists g i. Here's another way of saying it. So the sexual curvature of m i m g i is greater than or equal to let's say minus one divided by i where i goes to infinity. So this approach is zero getting closer and closer to zero. And let's say the diameter of m g i is a fixed number maybe pi. Okay. So this is why we call them almost non-negative and correct of course. And many folks in this class have not been studied that much but I think they really are important for understanding the whole picture and for understanding in general manifolds with lower curvature bound. Okay. Just like in the case of manifolds with bounded curvature, where almost flat manifolds play a central role in the work of Chica Foucaille and Gromov. Okay. So that's expected but hasn't been really developed. All right. By the way in this class any manifold in this class is also finally covered by a compact manifold that fibers over a nil manifold with fiber being simply connected. Okay. But in general the class of fundamental group is larger. But if we move on to assume that m is simply connected then there's not a single example known that of even going all the way to almost non-negative curvature that couldn't possibly have positive curvature. No obstructions are known. Okay. Let me add so this is so what are the other general theorems about positive curvature? They also go back a long time. They go back to synths or sing. So it's also about just about the fundamental group. If m is even dimensional this implies that pi1 is either trivial or z mod 2. This is when m is orientable and this is of course when it's not. Okay. And if it is odd dimensional then m is orientable. And these are results going back to synth in the 30s. Okay. And all of them are sort of like baby Morse theory in the sense. You look at the minimum general length of general minimal shortest closed deodesic or whatever in the homotopic class and you show that it has to be trivial. Okay. So this is just a simple variation formula. So in this example here you take a suppose m is not simply connected. You take a shortest closed deodesic that's non-trivial. You find from the assumptions a parallel field and you just look at the variation in that direction and then you find shorter curves. Okay. There are several theorems like this and I actually all put them in the same block. I kind them synth type theorems. Okay. There's a theorem due to Frankel that says in positive curvature if you have two totally deodesic sub manifolds they intersect if their dimensions are too big. It's exactly the same proof kind of going back to Weinstein if you have an isometry and such a manifold you find fixed points in even the mentions if it preserves orientation. And that by the way is very similar to Berger's theorem about zeros of killing fields. Okay. So he proved that any killing field on an even dimensional manifold has a zero and in order mentions it's true that if you have two killing fields they're dependent at one point. Okay. So these are the general results that I know of for manifolds of positive curvature without any further assumptions. Okay. So that's end of story in that sense. Okay. There's much more work that has been done but all of it with additional assumptions. So for example what what Marcel did of course was to prove the quarter pinching theorem and he proved other things very interesting things positive curvature. But these are not like completely general results. Okay. So I'm trying to stick with completely general results at this point and see what what else one can do. All right. So now I want to move on to some basic conjectures. And this is really not working well. Okay. Okay. Can you see it? Okay. I know the theorem so I don't need to see it. Okay. I can't see it. All right. So what are some main conjectures? Maybe I'll give up and just put this one here and then then do it reverse afterwards. Okay. So we can't avoid Hopf and there was mentioned by Bruce earlier today. So the Hopf has two basic conjectures attributed to Heinz-Hopf. One of them is that the Euler characteristic of M is greater than or equal to zero when M is non-negatively curved and actually it should be positive when M has positive sectional curvature. And the other one, the one that Bruce mentioned is the question whether S2 cross S2 is in or not. I guess if it's a conjecture, the conjecture was if it's not, but maybe Hopf asked just asked the question. So, okay. Now this has two very natural more general related questions I would say. So one is the following. If you take two manifolds N and V and take their product, there is not a single such example known to have positive curvature. I'm not saying conduct time anything. I'm just saying there's not such an example known yet. So of course, here's what we're talking about as 2 cross S2. Well, I mean this is not, if I take them to be simply connected, okay, take them to be simply connected. So only Rp2 cross Rp2 doesn't have positive curvature. So the zoom N and V is simply connected. And the other direction is how about you could think about symmetric spaces and I'll use, I guess this goes back to best this notation maybe, I think so, right. Compact rank one symmetric space, okay. So S2 cross S2 has ranked two, okay. And you could ask this question for any symmetric space a rank at least two. Is there a not right. Plus I'm saying it shouldn't have positive curvature. Oh, yes. Oh, then I can't attribute this. I just wanted to do that. Symmetric space, darn. Symmetric space, thank you. Of rank greater than or equal to two. Yes, thank you. Of course, that's what I meant. Yes, the other ones do. All right, so these are basic questions and here's a basic conjecture due to bot. That says that the, if we look at the, if M is also in this class, then the sequence of bedding numbers of the loop space of M grows at most polynomially. Okay. There's, there's a, this is, I'm not sure this is the formulation bot had because it's not in any, it's not written down, but bot wrote a paper with the Berger way where they looked at pinched manifold. So in this class with two bounds of the curvature and they provided estimates for the bedding numbers, but they're way from this way off from there and the exponential growth. They are estimates in terms of the pinching, but they have nothing to do with this conclusion that you would like. Okay. Now, if you focus, so you could also, another question is, are you thinking of any field like in Gromov's theorem or what not? Okay. But if you, if you, so here we could say F and this could be any field, maybe. That's a reasonable conjecture, but it becomes really obviously interesting in particular. There's a lot of consequences. If you take F to be the rational numbers, because then this condition is actually equivalent to the following that the dimension of the homotropic groups, the rational homotropic groups is finite. Okay. So this follows via the minimal model theory of Solomon and so on and so forth. This is old also, right? This is also way past. Okay. Now, this by itself, I'll just write down, it has lots of consequences that have been proved, topologically. One of them is that the Euler characteristic is then greater than or equal to zero. This follows from this. Okay. So, so the butt conjecture would imply one part of the Huff conjecture at least. The other part is that it's, and so, so which suggests what to try to do if you want to prove the Huff conjecture, if you believe butts. So, because being positive for the Euler characteristic is then equivalent to all odd rational better numbers to be zero. Okay. So in a sense, you might as well just prove that to prove the Huff conjecture. All right. So, all right. Another consequence is related to Gromov's theorem, and it is that the sum of better numbers, rational better numbers here, is at most two to the end where n is the dimension. See, that's a good bound, right? And in general, to distinguish these classes, one might want to prove just bounds that would allow you to distinguish them. That's what maybe one possibility, but that's, that's really all I wanted to mention here. Okay. Now, so, given this bleak situation about general results, I'd like to provide some other bleak news about how many examples do we know of this? Okay. So examples, and constructions. So what is, what are, how do we construct examples? Well, the, so let me talk about some constructions first. So if you start with a manifold m in m zero, and we look at, at, I'll call these projections. So you look at, at, at a, like a Croson map, m to n pi, where this is a Riemannian submersion, then this implies that this guy is also in this class, and you might be clever or lucky or whatever, and show that sometimes it is in this class. Okay. That's a very important construction. Now, if you look at, if you look at this class, Tom and I went out together last time, I'm losing my wise two now. If you look at this class, then you can take products, right? Products in this class work, of course. And then there's the bundles, and I would say in this class, what do I mean by bundles? It's the, that's the converse of this one in a certain sense. This is a very hard direction to go in. So basically, you start with something, you start with, with something, a base in a class, and you have a bundle over it, maybe a principal bundle, maybe a principal G bundle. I'll just stick to that right now. Then it follows that when B is in this class, that P is also in this class. So this is a way to construct many examples of almost non-negative curve manifolds that doesn't work in either of these categories in general, right? That's why this inverse, the converse of the sole theorem is very hard, right? So to go up to bundles is very difficult, right, to construct metrics. So these are some of the constructions. There's one other construction, and this is called the special gluings. And Jeff was the first to make such interesting constructions, namely, for example, and now, now I can use my word cross, okay? So if you take m and n, okay, compact rank one symmetric spaces, then Cheeg approved that their connected sum has a metric of non-negative curvature, okay? Basically using the, if you cut out a point from one of these space, you have a bundle and you change it so that it becomes product near the, near the boundary so that you can glue them together isometrically with a, by means of unit sphere. And there are other constructions about gluings that have to related to, to manifolds of core motifs one that Wolfgang Tiller and I developed, but let me just move on. So these are the examples and they have led to the, to the follow, these are constructions that led to the following examples. So to get started, you need, you need something, okay, I'll not write the main source for examples to this day, just leagrups, compact leagrups. Or leagrups with binary metrics, because they, if you take a binary metric, they belong to this class, okay? So therefore you can, you can, you can do all these constructions with them, okay? So you can take quotients. So all homogeneous spaces have metrics of non-negative curvature. The ones with positive curvature have been classified by Birard Bergieri and, and actually, so the examples constructed started with Marcel's work on the so-called normal homogeneous spaces. There are three of those, two in dimension seven, one dimension 13. And then Wallach constructed, classified the even dimensional examples in dimensions six, seven and 24. I have no idea if that 24 has to do with Brentelsenau, but I don't think so. But, and then Alaph and Wallach constructed a whole infinite family in dimension seven, okay? And that's it, all right? So, so, so if you look at homogeneous examples from this class, there are of course the crosses that were always there. And then they're the ones you have to work for to construct. And they exist in dimensions, I'll just write down the dimensions six, seven, 12, 13 and 24. 16 is one of the crosses, the Cayley plane kind of also unusual. But there is one example here, it's a flag manifold, one exam, another flag manifold, one exam, another flag manifold. This is a Berzei's example in dimension 13. And then there are infinite many here. Okay? Taking quotients, you can also take what's called bicochons, okay? So, so these are denoted. I really would like not to move on to another board. So, so a group eights can act on both left and right. So, we have a subgroup of d cross g acting on itself, maybe freely. And then these also have all have metric of non-negative curvature, of course. And there you get many more examples now with positive curvature. Okay? So, you get more examples in dimension seven and dimension 13. These are due to Esenberg, you have infinitely many. And here, due to Berzeichen, you have infinitely many examples. And that was where the situation stood up until recently. So, so these are, these are the constructions and examples that you have at your disposal. Of course, in non-negative curvature, you can do a lot more, right? You can take products, you can take quotients and this and that. So, that's why you get so many more examples. Now, so let me move on to see what, what you might do since, since if you're really not able to prove, let's say, Bruce is suggesting that you can find positive curve metrics on this two cross two arbitrary cross to the product metric or anything else, approve that hop the answer is no, you can't find such a metric or anything else. You've got to do something, right? And various approach, you could look at the pinching problem, pinch curvature, etc. It doesn't really need lead to new ideas, I would say. You can even look at the pinching of the size in terms of the diameter. Does, does also not lead to really new insights and new examples or anything. So, what you can do is, you can study all these examples and see if you get new ideas about manifold to positive curves or non-negative curves or whatnot. Or you could try to find more. Okay. One way to try to find more is to say, let's try to start over again and say, let's admit it. We can't do this in general yet. Let's just see what we can do in the presence of symmetries. So, like Bruce explained in dimension four, if you have a circle of isometries, you can actually prove what the manifold is. The manifold, if it has non-negative curvature, it's either S4, Cp2, S2 cross S2 connected some Cp2 plus minus Cp2. And you even know the action up to a very different morphism, complete classification. Okay. And so, you might, you might want, wonder what you can do in general. So, let me just write that down here. So, let me say, let's study the classes. m plus, n zero, and m zero minus in the presence of large or special isometric group actions. And it's not completely unreasonable to do it. Certainly, Broward's insight in dimension four, but all the known examples have a fair amount of symmetry, all of them, right? So, you could ask, is that how much does it take to classify? And it actually might give you new ideas to construct new examples as well. Okay. And this indeed has happened. So, let me just give you two examples of a notion of large. So, what could it mean that if a group G acts on m, that this group G is large? Well, one knows this obviously that the dimension of G is large relative to the dimension of m. This is called the degree of symmetry. Also studied by Wu Yixiang. I'd studied topologically by many people. You could think about the rank of G being large. So, you take a maximal torus. I just think of torus actions that play a role also in collapse manifold with bounded curvature. So, torus groups are certainly important. Another way of thinking of large is if the so-called core homogeneity, which is the dimension of the manifold, marginal of the group, so the orbit space, that this one is small. Okay. So, I will, oops, I will tell you about some results. Let's see. What should I do? I think I should bring back since we all, oh, I want that one down. Maybe now I'm really not so fast. Okay. All right. So, I will focus on just the rank condition. So, this one. So, suppose you have a torus that acts on a manifold m, isometrically. And here, in this situation, I'll assume m has positive curvature. And what you can show, for example, is in this case, is that the dimension of k cannot be too big. It can be in most roughly half of the dimension. And if it is, you can tell what it is, the manifold. It's actually like a cross, okay, up to diffeomorphism. Now, Wilking showed that you can do better. You can go down to about k is about one fourth of the dimension of m. And you can still say what the manifold is. It's rough again, like a cross, up to tangential homotopy equivalence. So, this equivalence is up to tangential homotopy equivalence. But there are also ways to address these questions due to Hopf in this setting. And Lee cannot, has a number of results related to this. Namely, answering basically, addressing the Hopf's questions. Forget there's two crosses, two. But think about the other ones for positive curvature. So, with k growing logarithmically as a function of n, as a logarithm function of n, there are many, there are many results. He has several results showing that then you cannot have a product, m not a product, and m, the Euler characteristic must be positive. So, what does this mean? So, of course, in this case, if here k grows linearly, it's like one, one fourth of the dimension, grows linearly, right? And then you can say a lot. You can say what the Betty numbers are and everything, right? So, in particular, this is true and they're also not products, right? That they are just like crosses, okay? But if you have much slower growth, in, so this is interesting at high dimensions, of course, in particular, then you could still conclude that you can't have manifolds with this amount of symmetry, torus symmetry, and, and m being a product of two manifolds, or examples like this. And also, you can conclude that the Euler characteristic is positive, okay? Now, here's the most spectacular work in this direction that just happened very recently, at least in my mind. So, so this is joint work due to Kenneth and Wilking. So, it simply says the following. If m is in this class, simply connect it, and t5 acts on m isometrically, then the Euler characteristic of m is at least two. So, this is, this is an amazing result in the sense that this is, doesn't matter what the dimension of m is, right? So, it has some symmetry, but it does, it tells, no matter how big dimensional manifold or positive courage you're looking at, it's one of those with positive Euler characteristics, and, and they can do more, apparently, than this, okay? This is proved in studying the fixed point sets of this and the topology of the fixed point sets. And so, this is, this is really something, okay? How am I, what am I supposed to stop? You can go a little further, but I think about, okay, so I will skip one part, but let me just say, let me just add then that, so in the other, in the other direction, and with the other notion of large, large points, they say the co-homogeneity. Let me, okay, let me just state the result in words. So, again, Wilking has a, has a, has a fundamental result that says if, if you fix the co-homogeneity to k, no matter what it is, could be 10 or whatever you want, then if the dimension is large enough, and it's not like one of these Gromov numbers, then the manifold is, is of the type of a cross, okay? If it has that co-homogeneity, okay? So, for example, look at the homogenous that are gone somewhere. How many, did, they dried up at dimensions 24, okay? And beyond that, they're only the crosses, okay? So, in co-homogeneity one, Zilla, and Wilking proved, gave sort of an, an exhaustive class of possibilities that is a classification in all dimensions, but seven, okay? In all dimensions, but seven, the result is the following. They are either across a normal homogenous space, or there are some infinite family of bzikin examples, and we know exactly what they are, okay? In dimensions seven, there are the, there are the two normal homogenous spaces. These are the only homogenous ones. And then there's an infinite family of the Eschenberg examples that also are co-homed to H1. And then there are two infinite families of candidates for positive curvature. That was what I was sort of driving at, right? And so, so in co-homogeneity one, they're like pk, qk candidates of new manifolds of positive curvature. I should say a means of that p1 is the seventh sphere, so this doesn't count. And q1 is one, the first alfamilic example, I call it 1, 1. The normal homogenous one actually. But p2 is known to be, now is in this class. This was proved by Derrick Hart, and independently by myself Verdiani and Ziller. By the way, Verdiani did the even-dimensional classified case before we did the odd-dimensional classification of these. Very recently, Derrick Hart has claimed that he can also do p3, q2, and q3. So these are new examples of manifolds of positive curvature, and so it sort of validates at least that this is not a bad approach to try to find new examples. And not even to speak of co-homogeneity two or whatnot. And I had meant to talk about co-homogeneity two, and the only reason was that there you can prove that if you are co-homogeneity two and in this last class, then you satisfy the bot conjecture, okay, then you're rasterly elliptic. I also would like to point out, and we'll finish by that. So if we go back to this list of examples, co-homogeneity zero means homogenous, we already saw that all the ones, they're all in M0, okay, I don't need to think about the almost non-negative curve. Co-homogeneity one, these are three distinct classes, okay, we have this very strong restriction to positive curvature. There are lots of examples that have non-zero curvature, but there are lots of examples also that do not have non-negative curvature, okay, but they all have almost non-negative curvature, right. In co-homogeneity two, most do not have almost non-negative curvature, period, but the ones that do, you can analyze, and in particular, they do, they are rasterly elliptic, okay, so let me stop here. Questions? From now, at least one question. What else is in the future? Yeah, well, I didn't have time. Well, what I, what I, I had a couple of things I want, so first of all, there's this question about why are only these crosses in high dimensions, what's going on? Maybe there's something to it, okay, or is it just because we are too ignorant and construct, all examples don't know how to construct examples other than using Lie groups, okay. Now, from, from that point of view, I think, so, so there's a lot of constructions now, we know much more now about a very interesting generalization of group axons, namely so-called singular Riemannian foliations, so these are locally everywhere equidistant, I mean, leaves, but they don't have to have the same dimension, but they are locally everywhere equidistant, these are called singular Riemannian foliations, and there's an abundance, no, Marco Rades has done a lot of really nice work on unit spheres, for example, not at all homogeneous, and so there's got to be a lot of examples of these in other, on other manifolds, right, and, and this recent work of Martin Kering, Chris Lanshanca and Gertze that showed that all exotic seven spheres, now we know that all exotic seven spheres have metrics of non-negative curvature, and this uses what Wolfgang and I did, but they took quotients, and one way of looking at the quotients that they looked at these constructions were they, that you can think of them as, as the co-homony to one picture with like a singular Riemannian foliation, where all the principal leaves are bicrotions instead of being homogeneous spaces, okay, so this is sort of like a hybrid between really general singular Riemannian foliations and homogeneous ones, right, but it shows that there are definitely interesting ways to think about these items in, in this context of singular Riemannian foliations, so, so I, I think that maybe one could construct examples that way, another one was there's a big old problem in, from the beginning of Alexandrov geometry that says that, that the boundary of an Alexandrov space should be an Alexandrov space with the same lower curvature bound, that's completely open, but recently Peterson and I and a student of mine, Adam Moreno, showed that it's true for leaf spaces of singular Riemannian foliations, so maybe another source is to look at all these general objects, take the boundary of such a thing, has non-negative curvature maybe, maybe positive curvature, in some cases they are manifold, maybe you can do deformations and so on and so forth, so those were some of the things I wanted to talk about, yeah. Well, if there are no further questions, thank you very much.
results that led to the proof of Topologov's and for understanding in general manifolds with positive when M has positive sectional curvature. metrics. So these are some of the constructions.