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?

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?