Can one recover the smooth Gauss Bonnet theorem from the combinatorial Gauss Bonnet theorem as an appropriate limit?

Question

First let me state two known theorems.

Theorem 1 (for smooth manifolds): Let $(M,g)$ be a smooth compact two dimensional Riemannian manifold. Then $$ \int \frac{K}{2 \pi} dA = \chi (M) $$ where $K$ is the Gaussian curvature, $dA$ is the area form and $\chi(M)$ is the Euler characteristic.

Theorem 2 (combinatorial version): Let $M$ be a two dimensional simplicial complex, with vertices, edges and faces (essentially a bunch of triangles glued together along the edges). Define the function $K:M \rightarrow \mathbb{R}$ to be zero at any point that is not a vertex. At any vertex sum up all the angles at the vertex and look at the deviation from $2 \pi$. This is the value of $K$ at a vertex. Then $$ \sum_{p\in M} \frac{K(p)}{2 \pi} = \chi(M) . $$

Both these statements are known as Gauss Bonnet Theorem. My question is the following: Is it possible to use Theorem 2 in some way to prove Theorem 1? In other words can one recover Theorem 1 from Theorem 2 as some sort of an appropriate "limit"?

A second question is: Is there are a more general theorem from which both theorem 1 and theorem 2 arise as special cases? Probably some version of the theorem where $K$ simply has to be a measurable function?

Answer 1

The answer is yes, to both questions.

First question first. For any geodesic $n$-gon $P$ on $M$, i.e., a simply connected region of $M$ whose boundary consists of $n$-geodesic arcs, define

$$ \delta(P)= \mbox{sum of the angles of $T$}-(n-2)\pi. $$

Note that if $M$ were flat, then the defect $\delta(P)$ would be $0$. This quantity has a remarkable property: if $P= P'\cup P''$, where $P, P', P''$ are geodesic polygons, then

$$ \delta(P)=\delta(P')+\delta(P'')-\delta(P'\cap P'') $$

which shows that $\delta$ behaves like a finitely additive measure. It can be extended to a countably additive measure on $M$, and as such, it turns out to be absolutely continuous with respect to the the volume measure $dV_g$ defined by the Riemann metric $g$. $\newcommand{\bR}{\mathbb{R}}$ Thus we can find a function $\rho: M\to \bR$ such that

$$\delta= \rho dV_g. $$

More concretely for any $p\in M$ we have

$$\rho(p) =\lim_{P\searrow p} \frac{\delta(P)}{{\rm area}\;(P)}, $$

where the limit is taken over geodesic polygons $P$ that shrink down to the point $p$. In fact

$$ \rho(p) = K(p). $$

Now observe that if we have a geodesic triangulation$\newcommand{\eT}{\mathscr{T}}$ $\eT$ of $M$, the combinatorial Gauss-Bonnet formula reads

$$\sum_{T\in\eT} \delta(T)=2\pi \chi(M). $$

On the other hand

$$\delta(T) =\int_T \rho(p) dV_g(p), $$

and we deduce

$$ \int_M \rho(p) dV_g(g)=\sum_{T\in \eT}\int_T \rho(p) dV_g(p) =\sum_{T\in\eT} \delta(T)=2\pi \chi(M). $$

For more details see these notes for a talk I gave to first year grad students a while back.

As for the second question, perhaps the most general version of Gauss-Bonnet uses the concept of normal cycle introduced by Joseph Fu.

This is a rather tricky and technical subject, which has an intuitive description. Here is roughly the idea.

To each compact and reasonably behaved subset $S\subset \bR^n$ one can associate an $(n-1)$-dimensional current $\newcommand{\bN}{\boldsymbol{N}}$ $\bN^S$ that lives in $\Sigma T\bR^n =$ the unit sphere bundle of the tangent bundle of $\bR^n$. Think of $\bN^S$ as oriented $(n-1)$-dimensional submanifold of $\Sigma T\bR^n$. The term reasonably behaved is quite generous because it includes all of the examples that you can produce in finite time (Cantor-like sets are excluded). For example, any compact, semialgebraic set is reasonably behaved.

How does $\bN^S$ look? For example, if $S$ is a submanifold, then $\bN^S$ is the unit sphere bundle of the normal bundle of $S\hookrightarrow \bR^n$.

If $S$ is a compact domain of $\bR^n$ with $C^2$-boundary, then $\bN^S$, as a subset of $\bR^N\times S^{n-1}$ can be identified with the graph of the Gauss map of $\partial S$, i.e. the map

$$\bR^n\supset \partial S\ni p\mapsto \nu(p)\in S^{n-1}, $$

where $\nu(p)$ denotes the unit-outer-normal to $\partial S$ at $p$.

More generally, for any $ S$, consider the tube of radius $\newcommand{\ve}{{\varepsilon}}$ around $S$

$$S_\ve= \bigl\lbrace x\in\bR^n;\;\; {\rm dist}\;(x, S)\leq \ve\;\bigr\rbrace. $$

For $\ve $ sufficiently small, $S_\ve$ is a compact domain with $C^2$-boundary (here I'm winging it a bit) and we can define $\bN^{S_\ve}$ as before. $\bN^{S_\ve}$ converges in an appropriate way to $\bN^{S}$ as $\ve\to 0$ so that for $\ve$ small, $\bN^{S_\ve}$ is a good approximation for $\bN^S$. Intuitively, $\bN^S$ is the graph of a (possibly non existent) Gauss-map.

If $S$ is a convex polyhedron $\bN^S$ is easy to visualize. In general $\bN^S$ satisfies a remarkable additivity

$$\bN^{S_1\cup S_2}= \bN^{S_1}+\bN^{S_2}-\bN^{S_1\cap S_2}. $$

In particular this leads to quite detailed description for $\bN^S$ for a triangulated space $S$.

Where does the Gauss-Bonnet formula come in? As observed by J. Fu, there are some canonical, $O(n)$-invariant, degree $(n-1)$ differential forms on $\Sigma T\bR^n$, $\omega_0,\dotsc, \omega_{n-1}$ with lots of properties, one being that for any compact reasonable subset $S$

$$\chi(S)=\int_{\bN^S}\omega_0. $$

The last equality contains as special cases the two formulae you included in your question.

I am aware that the last explanations may feel opaque at a first go, so I suggests some easier, friendlier sources.

For the normal cycle of simplicial complexes try these notes. For an exposition of Bernig's elegant approach to normal cycles try these notes.

Even these "friendly" expositions with a minimal amount technicalities could be taxing since they assume familiarity with many concepts.

Last, but not least, you should have a look at these REU notes on these subject. While the normal cycle does not appear, its shadow is all over the place in these beautifully written notes.

Also, see these notes for a minicourse on this topic that I gave a while back.

Answer 2

That should just be a comment but it's getting to long.

Also related is the theory of surfaces with bounded integral curvature developed in the 50's by A.D. Alexandrov and its followers. This class of surfaces contains both smooth and polyhedral surfaces.

To be more precise, in Alexandrov words a surface with bounded (integral) curvature is a compact topological surface $(X,d)$ with a geodesic distance $d$ such that for any finite collection of (nice) disjoint geodesic triangles $T_i$, $\sum |e(T_i)|<C$ for some $C$ indecent of the collection of triangles. Here the excess $e(T)$ of a triangle is "sum of angles of $T$ minus $2\pi$". (Actually I put quite a lot of stuff under the rug, you have to define what angles are in this context, and it's not clear they exist, so one consider upper angles instead).

From this one defines, essentially in the way Liviu did, a curvature measure $\omega$ on $M$, wich will be $K_gdv_g$ in the smooth case and $\sum K(p)\delta_p$ in the polyhedral case.

The two versions of Gauss Bonnet you gave can be rephrased as $\omega(M)=2\pi\chi(M)$, which in fact holds for all Alexandrov surfaces.

Moreover, one can deduce each one from the other by an approximation process. One can approach a smooth metric by a sequence of polyhedral metrics or a polyhedral metric by a sequence of smooth metrics in such a way such that the curvature measures with respect the approximating metrics (weakly) converge to the curvature measure of the approximated metric.

This material can be found in 'The intrinsic geometry of surfaces with bounded curvature' by A.D. Aleksandrov and Zalgaller.

Answer 3

This kind of thing has actually been very actively investigated, in the context of o-minimal structures, and also in rather applied contexts (of computational topology). Check out this nice blog post for more info and further references.