I'm now attending a reading seminar on the algebraic topology.


The seminar treats the book of Bott & Tu (Differential Forms in Algebraic Topology) and Milnor (Characteristic Classes).


In those books, theorems on the Riemannian manifolds are frequently just mentioned and used.


To mention some examples


1. Riemannian manifold has a good cover.
2. Exponential Map is used to find a tubular neighborhood for a pair of manifolds. (where one is a submanifold of the other ) and its properties are used in computations on the dual cohomology class and the diagonal cohomology class.
3. Argument in a proof which states that we can reduce the general case to a local open submanifold with the Euclidean standard metrics.


and maybe more.


When I browse books on differential geometry or Riemannian manifolds, I get the feeling that I cannot avoid studying the standard materials like the connection, tensors...


But I have no time to study all that materials.


Is there some shortcut to understand those materials (at least for good manifolds) without studying all the details of these differential materials? (Maybe is there some axiomatic approach?)


Any suggestions on the references are welcome.


Thank you very much!



Any shortcuts to understanding the properties of the Riemannian manifolds which are used in the books on algebraic topology
The proof of the existence of good covers is contained in Bott & Tu on pages 42-43, though that proof does refer out to Spivak (see below).


In general, if your main goal is to study (algebraic) topology of manifolds, you probably don't need to know much about metrics and connections and that sort of thing that typical differential geometry books spend a lot of time on. It sounds like what you really need is some material on differential topology. I don't know how much they'll cover of the specific things you're looking for, but here are a few suggestions to check out:


1. Topology and Geometry by Glen Bredon. This might be a particularly good book for you as it really combines the two topics pretty well.
2. Michael Spivak's A Comprehensive Introduction to Differential Geometry, Vol. 1. Despite the title, the first volume is more about differential topology than geometry. Also this is what Bott and Tu cite for their key fact needed for the existence of good covers.
3. Differential Topology by Guillemin and Pollack - this is a very readable introduction.
4. Introduction to Smooth Manifolds by John M. Lee - this is oriented a bit more toward geometry but you can find a lot in it.


I'd recommend skimming through these (and others) to find one that suits you and has the kinds of things you're looking for.


