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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, of Forman and also some wonderful work by Luo. 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:

Are there combinatorial analogues of curvature that satisfy Gauss-Bonnet for abstract simplicial complexes and Myer's theorem for triangulated complete Riemannian manifolds?

If the answer is "yes", then what are they? And if "no", then:

What are the popular combinatorial definitions of curvature different from the ones that show up in the papers mentioned above?

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It is easy to see what the Pfaffian of curvature should be for Gauss-Bonnet, since you can compute the Euler characteristic by locally adding up fractional pieces of $k$-cells. It is harder to see what matrix could have this Pfaffian. –  Will Sawin Jun 13 '12 at 22:21
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For a discrete analog of Ricci curvature, see here arxiv.org/abs/1103.4037 and references inside. They have no result similar Gauss-Bonnet, but still study other properties a notion of curvature for graphs should have. –  Camilo Sarmiento Jul 12 '12 at 16:55

3 Answers 3

up vote 4 down vote accepted

A combinatorial Gauss-Bonnet theorem is proved by B. Chen in THE GAUSS-BONNET FORMULA OF POLYTOPAL MANIFOLDS AND THE CHARACTERIZATION OF EMBEDDED GRAPHS WITH NONNEGATIVE CURVATURE (P. AMS, 2009), but previous work in the same vein was done by Schlafli and Poincare. More recently also by this same Bienfang Chen.

For Myers-Bonnet, see this preprint by Saucan

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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 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 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 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.

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Dear Liviu, thank you for this comprehensive answer, I will take a look at the references. Is the link to Fu's notes broken? –  Vidit Nanda Jul 12 '12 at 15:56
    
I fixed the link. Look at the notes about the normal cycle. –  Liviu Nicolaescu Jul 12 '12 at 16:22

My favorite combinatorial definition for curvature in an $n$-manifold $M$ (given as an abstract simplicial complex) is the angle defect around codimension-2 simplicies, measured using the PL-metric in which all edges have unit length. This makes the curvature at an $(n-2)$-simplex $\sigma$ depend only on the number of $n$-simplices in the manifold with $\sigma$ as a face (the degree of $\sigma$.) That is, $curv(\sigma) = 2\pi - \theta_n deg(\sigma)$ where $\theta_n = \cos^{-1}(1/n)$ is the dihedral angle in a regular $n$-simplex and $deg(\sigma)$ is the degree of $\sigma$.

Using angle defect at codimension-2 simplices goes back to Regge's work, T. Regge (1961). "General relativity without coordinates". Nuovo Cim. 19 (3): 558–571.

In http://www.combinatorics.org/ojs/index.php/eljc/article/view/v17i1r49 , I prove a Bonnet-Myers type theorem which gives sharp bounds on the edge-diameter of the 1-skeleton of $M$ under the assumption that $curv(\sigma)>0$ at every $(n-2)$-simplex $\sigma$ in $M$. Additionally, I prove a corresponding rigidity result (analogous to the rigid sphere theorems of Toponogov and Cheng) which shows that if such a manifold has the maximum allowed edge-diameter then it must be an $n$-sphere whose triangulation is (almost) completely fixed.

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