There appear to be many "combinatorial" definitions of curvature as applied to finite simplicial (or regular CW) complexes. For instance, we have the ideas of [Cheeger, Muller and Schrader](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-92/issue-3/On-the-curvature-of-piecewise-flat-spaces/cmp/1103940867.full), of [Forman](http://math.rice.edu/%7Eforman/ric.ps) and also some wonderful work by [Luo](https://www.ams.org/journals/era/2005-11-02/S1079-6762-05-00142-3/). This list just refers to work that I've come across over the last few years and should not be considered complete...


Of course, each definition appears to have a different "domain" to which it applies, and no two are equivalent on the intersection of their domains. This is in sharp contrast to the smooth curvature theory, some of which is even canonically introduced to undergraduates in standard calculus sequences.


Moreover, each theory mentioned above has inherent gaps in terms of establishing a good analogy with smooth curvature. Cheeger et. al work in a setting that is borderline non-combinatorial and focus on approximating curvature of smooth objects by piecewise flat ones "in-measure"; Forman does not require embedding but there is provably no version of Gauss-Bonnet for his definition of curvature. Luo's work restricts to triangulated 3 manifolds with boundary, etc.


It is not straightforward to come up with a wish list of what one desires from a definition of combinatorial curvature, but certainly: Gauss-Bonnet, Myer's theorem, and well-behavedness under cell subdivision come to mind.


So:



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> Are there combinatorial analogues of curvature that satisfy Gauss-Bonnet for abstract simplicial complexes and Myer's theorem for triangulated complete Riemannian manifolds?
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If the answer is "yes", then what are they? And if "no", then:



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> What are the popular combinatorial definitions of curvature different from the ones that show up in the papers mentioned above?
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Combinatorial analogues of curvature
Perhaps, the most general concept of curvature (measure) is that of *normal cycle*. This is an object $N^S$ naturally associated to a reasonably nice compact subset $\newcommand{\bR}{\mathbb{R}}$ $S\subset\bR^n$. PL sets are reasonabky nice and, more generally, the semialgebraic sets are nice. The compact smooth submanifolds of $S$ are also nice.


Briefly, $N^S$ is an $(n-1)$-dimensional current on $\bR^n\times S^{n-1}$. Here is briefly its definition.


If $S$ is a compact domain with $C^2$ boundary, then the outer normal to the boundary $\newcommand{\bn}{\boldsymbol{n}}$ defines the *Gauss map* $\newcommand{\pa}{\partial}$ $\bn:\pa S\to S^{n-1}$, and then $N^S$ is defined as the current of integration along the *graph of the Gauss map*. $\newcommand{\ve}{{\varepsilon}}$ If $S$ is not such a domain, but it is sufficiently nice, then for $\ve>0$ sufficiently small the compact set


$$S\_\ve:=\bigl\lbrace x;\;\; {\rm dist}(x, S)\leq \ve\bigr\rbrace $$


is a domain with $C^2$ boundary and one can show that as $\ve \searrow 0$ the currents $N^{S\_\ve}$ converge weakly to a closet current $N^S$ which is rather nice. (You can think of $N^S$ as being the current of integration along several smooth oriented manifolds with some multiplicities. ) In other words $N^S$ can be thought of as the graph of the Gauss map of a thin tubular neighborhood of $S$.


Where does curvature come in? In the smooth case, Gauss' Theorem shows that the Gauss map encodes the curvature, so it is not surprising that in this case $N^S$ has something to do with the curvature. How does it work in general?


On $S^{n-1}\times \bR^n$ there exist certain universal $(n-1)$-forms


$$ \alpha\_0,..., \alpha\_{n-1} \in \Omega^{n-1}(S^{n-1}\times \bR^n) $$


The integral of $\alpha\_k$ along $N^S$ is called the $k$-th *curvature measure* of $S$ and it is denoted by $\mu\_k(S)$. It can be expressed as an integral over $S$ of a certain measure, which may be singular if $S$ is. When $S$ is a smooth oriented submanifold of $\bR^n$, the measure describing $\mu\_0$ is given by the integration of the Pfaffian of the Riemann curvature. For PL sets this measure will have singular contributions coming from the angles of the faces, their areas etc. The general Gauss-Bonnet theorem states that if $S$ is a reasonably nice set, then $\mu\_0(S)$ is the Euler characteristic of $S$. The story is much more complicated and more beautiful than what I can squeeze in a few paragraphs, but I can give you some links to places where you can read more about this subject.


First, you should look at [these beautiful notes of J. Fu](http://alpha.math.uga.edu/%7Efu/research/research.html) who is one of the pioneers in this field. On his homepage you will find other beautiful things.


You can also look at [these notes of mine](https://www3.nd.edu/%7Elnicolae/morse-sing.pdf) where I give a description of $N^S$ for simplicial sets. There is a very close connection between the normal cycle and Morse theory and my notes discuss this aspect.


If you want a more elementary point of view on these curvature measure, then you should definitely have a look at [this REU project](https://www3.nd.edu/%7Elnicolae/val-simple.pdf) that I supervised a few years ago. I was very pleased with the final result of these three fine young people. This survey is very written with many figures and a lot of intuition. The last part deals with Morse theory and Gauss-Bonnet for two-dimensional simplicial complexes.


