A discrete Gauss-Bonnet type theorem

For a domain $D$ in the plane with smooth boundary $C$, the Gauss-Bonnet formula or Umlaufsatz $\int_{C}K(s)\;ds=2\pi\chi(D)$ relates the curvature $K(s)$ of the boundary curve with the Euler characteristic $\chi$ of the region. For a simply connected region $G$ for which the boundary is a simple closed curve, the total boundary curvature is $2\pi$. This Gauss-Bonnet type result is a form of Hopf’s Umlaufsatz and relates a differential geometric quantity, the boundary curvature, with a topological invariant, the Euler characteristic. In differential geometry, curvature needs a differentiable structure, while Euler characteristic does not. It is the transcending property between different mathematical branches which makes Gauss-Bonnet type results interesting.

We prove here a discrete version of a ”Hopf Umlaufsatz” [3] which is of combinatorial nature; curvature is an integer. The result applies to special two dimensional graphs which are part of a flat two dimensional background graph $X$, where the dimensionality is defined inductively. While the Euler characteristic is a topological notion, we need ”smoothness assumptions” to equate the total boundary curvature with the Euler characteristic.

The curvature, we consider here is $K(p)=2|S_{1}|-|S_{2}|$, where $S_{r}(p)$ is the arc length of the sphere $S_{r}(p)$ at the point $p$. The sphere $S_{r}$ is a subgraph of $G$ with vertices of all points of distance $r$ and edges consisting of pairs $(q,q^{\prime})$ in $S_{r}(p)$ such that $q$ and $q$ have distance $1$. As we will explore elsewhere, for many compact two-dimensional graphs $G$ without boundary, like triangularizations of polyhedra with 5 or 6 faces, the integral of the curvature $K(p)=2|S_{1}|-|S_{2}|$ over the entire graph is $60\chi(D)$. The ”smoothness” assumptions are more subtle than corresponding results for $K_{1}(p)=6-|S_{1}|$. For the later, the result $\sum_{p\in G}K_{1}(p)=\chi(G)$ is essentially a reformulation of Euler’s formula and holds for any ”two dimensional graph” with or without boundary. We will look at a relation between the ”first order curvature” $K_{1}$ and second order curvature $K$ at the end of this article.

The main result in this paper is the formula $\sum_{p\in\delta G}K(p)=12\chi(G)$ which holds for discrete domains $G$ and for a second order curvature $K$. To do so, we need to specify precisely what a ”smooth domain” is.

The background lattice $X$ plays here the role of the two-dimensional plane. Its vertices can be realized as the set of points $\{k(1,0)+l(1,\sqrt{3})/2\;|\;k,l\;{\rm integers}\;\}$. The edges are formed by the set of pairs for which the Euclidean distance is $1$. In the infinite graph $X$, every point $p$ has 6 neighbors. Together with edges formed by neighboring vertices, these points form the unit sphere $S_{1}(p)$, a subgraph of $X$. Similarly, any sphere $S_{2}(p)$ of radius $2$ in this discrete plane has length $|S_{2}|=12$. The curvature $K=2|S_{1}|-|S_{2}|$ is zero at every point of the background lattice $X$.

Figure: Curvature computation. The numbers near each vertex indicate the curvature of the point. At each of a few chosen points, we have drawn the spheres of radius $1$ and $2$ in $G$. Adding up the curvatures over the boundary gives 12. If a point has as a neighborhood a disc of radius $2$, the curvature is zero.

A finite subset $G$ of the triangular lattice $X$ defines a graph $(V,E)$, where $V\subset X$ is the set of vertices in $G$ and where $E$ is a subset of edges $(p,q)$ in $X$, pairs in $X$ have distance $1$ within $X$. We start by defining a dimension for graphs. To our best knowledge, this notion seems not yet have appeared, even so in the graph theory literature, several notions of dimension exist. The definition of dimension is inductive and rather general and does not require the graph to be a subset of $X$.

The dimension does not need to be defined. For example, a point which has a sphere which contains of one and zero dimensional components has no dimension. One could define inductively a fractional dimension by adding $1$ to the average fractional dimensions of the points on the unit sphere.

As an illustration of the notion of dimension, lets look at the platonic solids as graphs. The cube and the dodecahedron are one dimensional. The isocahedron and octahedron are two dimensional. The tetrahedron is three dimensional, because the unit sphere of each point is 2 dimensional. The cube and dodecahedron become two dimensional after kising (stellating) their faces. The tetrahedron becomes 2 dimensional after truncating corners.

For an interior point, the sphere $S_{1}(p)$ is a closed circle, for a boundary point, the sphere $S_{1}(p)$ is a union of one-dimensional arcs.

Remarks:
a) The set of subsets $\{A\subset{\rm int}(G)\;\}\cup\{G\;\}\;$ defines a topology on $G$ such that the interior of $G$ is open and the boundary $\delta G$ is closed.
b) The interior of a two dimensional graph $G$ is not necessarily a 2 dimensional graph. The disc of radius $1$ in $X$ for example is has a single interior point so that the interior is zero-dimensional.
c) Two dimensionality of a graph has no relation with ”being planar”. There are planar graphs like the tetrahedral graph which is three dimensional in our sense but which is planar. And there are graphs like triangularizations of a torus, which are two dimensional but not planar.
d) Topologically, one can show that the triangular graph $X$ is the only simply-connected two-dimensional flat graph without boundary.

The conditions (i),(ii), (iii) are natural. Condition (iv) assures that no unnatural ”fissures” can exist. Condition (v) assures that the connectivity topology of the domain and the connectivity topology of the interior set are the same.

Remarks:
a) We could additionally require the interior of a domain to be two-dimensional but we do not need that. Actually, the proof of the main theorem becomes simpler if we do not make this assumption. It would just lift a difficulty on a different level. For us it will be important to look at the dimension of points in the interior of $G$.
b) Some of these conditions for ”domains” have analogues in the continuum, where they are necessary for the classical Gauss-Bonnet to be true: we can not have ”hairs” sticking out of the domain for example. The closure of the complement of a domain is a domain too and we can not just leave out part of the boundary. Also in the continuum, it should not happen that parts of domains are tangent to each other. We also can not allow the boundary to be two-dimensional, like for the Mandelbrot set.
c) For a smooth domain, we can look at the interior $H^{\prime}={\rm int}(G^{\prime})$ of the complement $G^{\prime}$ of ${\rm int}(G)$. Then, the boundaries satisfy $\delta(G)=\delta(G^{\prime})$. The three sets ${\rm int}(G^{\prime})$, ${\rm int}(G)$ and $\delta(G)=\delta(G^{\prime})$ partition the graph $X$.

Figure: Examples of domains. The number in the upper right corner is the total boundary curvature of the domain.

Figure: Examples of graphs which are not domains. To the left, a set with 2, 1 and 0 dimensional points. It violates conditions (i). The second example is a set with both 2 and 1 dimensional points.

Figure: The left example is a two-dimensional set with no interior points and no boundary points. It violates condition (ii). The second one violates (v).

Figure: Examples of graphs which are not domains. The first violates (v), the third violates (iii).

Figure: Domains which are not smooth domains. The curvature of the first was computed while assuming the nearest neighbor connection to be an edge as required by condition (iv). If the connection is not in place (violating (iv)), the total curvature would be 24.

The following lemma allows us to deal more efficiently with eligible regions and eliminates many subsets which are not regions. It says that the set of interior points determines the region as well as its boundary.

Remark: For a simply connected region, also the boundary of $G$ determines the region, but we do not need that.

Remarks.
a) Note that $|S_{1}|$ is not necessarily the number of vertices in $S_{1}$. Similarly, $|S_{2}|$ is the number of edges in $S_{2}$ which is not always equal to the number of vertices in $S_{2}$.
b) The sphere $|S_{1}|$ does not necessarily have to be connected, nor does it have to have a defined dimension. It could be a union of a segment and a point for example.

Remarks.
a) This definition is motivated by differential geometry since one can derive an analogue formulas in the continuum $K=\lim_{r\to 0}\frac{2|S_{r}|-|S_{2r}|}{2\pi r^{3}}$ for a point on the boundary curvature of a region.
b) Note that as defined, $S_{2}(p)$ refers to the geodesic circle of radius $2$ in $G$ and not in $X$ so that every point $q\in G$ of distance $2$ in $X$ to $p$ belongs to $S_{2}(p)$ whether there is a connection within $G$ from $p$ to $q$ or not. The reason for this choice is that we do want the curvature definition to be nonlocal. This subtlety will not matter since for the definition of smooth curve, we anyhow disallow situations where points have a large distance within $G$ but small distance in $X$.

Figure: The first picture is a smooth domain. It is not simply connected although the two parts $S_{1},S_{2}$ of the regions are at first separated enough to get a curvature 24. In the second case $S_{2}$ ”feels” part of the other region $S_{1}$ and the curvature is not a multiple of $12$.

Figure: In the first picture, the domain is still not smooth because the complement is not a domain. The last example is a smooth domain. It has become simply connected.

Our main result of this paper is a discrete version of the ”Umlaufsatz”. It will be generalized to more general domains below.

To generalize the Umlaufsatz to domains which are not necessarily simply connected we first define the Euler characteristic of a region using Euler’s formula:

Example: for a simply connected region, the Euler characteristic is $1$.

Figure: A region, where no interior point can be removed any more and which has more than one interior point is not simply connected.

Remark. The Euler characteristic of ${\rm int}(G)$ and $G$ is the same if $G$ is a smooth region.

We could use the same pruning-etching technique as before. However, pruning and etching can lead to final situations which have no end points like a ring. Instead of classifying all these final situations, it is easier to reduce the general situation to a simply connected situation.

There are two ways, how to change the topology:

• •

  build bridges between different connected components.

• •

  fill holes to make the region simply connected

Merging different unconnected components is no problem. As long as their complement is a smooth region too, both the Euler characteristic as well as the total curvature add up.

1. We can assume the region to be connected, because both curvature as well as Euler characteristic are additive with respect to adding disjoint domains. To illustrate this more, we can also join two separated regions along with a one dimensional bridge. The curvature drops by $12$, the number of connected components drops by $1$.

Figure: Joining two regions changes the total curvature by 12.

By the Umlaufsatz for simply connected regions, the hole has total curvature 12. A key observation is that the point-wise curvatures at the inner boundary of $G$ enclosing the hole are just the negative of the corresponding point-wise curvatures of the hole. This follows almost from the definition of curvature and the fact that the circles $|S_{1}(p)\cap W|+|S_{1}(p)\cap G|=6$ and $|S_{2}(p)\cap W|+|S_{2}(p)\cap G|=12$ so that $K_{W}(p)+K_{G}(p)=0$.

Figure: Filling a simply connected hole from a larger region adds to the curvature exactly the same amount than the total curvature of the hole. The reason is that the point-wise curvatures of the removed inside region matches exactly the curvatures of the inner outside region if the inside region and the outside region have a common one-dimensional boundary.

This shows that if we fill a hole, the total curvature increases by $12$. Simultaneously, the Euler characteristic increases by $1$.

Alternatively, we could also cut rings:

Figure: When cutting a ring, the curvature changes from 0 to 12. Only the last region is a smooth region. The second last is a region but not smooth because the complement is not a region.

If we introduce identifications in the hexagonal background graph $X$, the topology of the background space changes. Identifying points along two parallel lines for example produces a flat cylinder. With a triangular tiling, we can tessellate a torus. Because there is no boundary now, the sum of the curvatures is zero, which is the Euler characteristic. Note that there are many different non-isometric graphs which lead to such tori. We call them twisted tori. As graphs they are different even if the number of faces, edges and vertices are fixed.

Figure: A flat torus obtained by identifying opposite sides of a rectangular domain in a hex lattice. The total curvature is zero.

The notion of regular domain can be carried over to discrete manifolds like the twisted tori just mentioned. Let $X$ be such a twisted background torus. We assume that it is large enough so that $S_{2}(p)$ is a circle at every point. Let $G$ be a subgraph of $X$ defined as before. We still have:

Remark: More flat compact graphs can be obtained using ”worm hole” constructions. Let $X$ be a possibly twisted torus as defined above and let $p$,$q$ be two points for which the discs $B_{r}(p)$ and $B_{r}(q)$ are disjoint and the spheres $S_{r}(p)$ and $S_{r}(q)$ are circles. Any orientable graph isomorphism between $S_{r}(p)$ and $S_{r}(q)$ produces an identification of points in $G=X\setminus B_{r-1}(p)\cup S_{r}(p)\setminus B_{r-1}(q)\cup S_{r}(q)$. Without identification, the total curvature of the boundary of the domain $G$ is $\sum_{p\in C}K(p)=12\chi(G)=-24$ because every removed disc produces curvature $-12$.

In this section, we consider a more elementary Gauss-Bonnet formula. The curvature is again defined by a Puisaux discretization, but only circles of length $1$ appear in the definition. It is a first order curvature.

For this combinatorial curvature, Gauss-Bonnet is much easier. For subgraphs of hexagonal lattice the boundary curvature is almost trivially equal to $6$ because the curvature is related to angles of the corresponding polygon: for a boundary point, $K_{1}(p)\pi/3$ is the interior angle of the polygon. Because $\sum_{p}K_{1}(p)\pi/3=2\pi$ by the polygonal version of the Umlaufsatz, we have $\sum_{p}K_{1}(p)=6$.

Actually, Gauss-Bonnet holds in great generally for arbitrary two-dimensional graphs with or without boundary. Since it is so closely related to the Euler characteristic, we should the attribute it to Euler, even so we are not aware that Euler considered $K_{1}(g)$, nor that he looked at the dimension of a graph.

Remarks.
1) This result does not need the rigid requirements on the ”domain” as before nor does the graph have to be part of $X$; it works for any two-dimensional graph, with or without boundary.
2) The result appears in a different formulation, which does not make its Gauss-Bonnet nature evident: the Princeton Companion to Mathematics [2] mentions on page 832 the formula $\sum_{n}(6-n)f_{n}=12$, where $f_{n}$ is the number $n$-hedral faces and summation is over all faces. This is an equivalent formulation, but it makes the Gauss-Bonnet character less evident.

Note that the combinatorial Gauss-Bonnet theorem is entirely graph theoretical. It avoids the pitfalls with the definition, what a polyhedron is [4]. (Common definitions of ”polyhedra” refer to an ambient Euclidean space or impose additional structure on a graph). We can take a general finite graph which is two-dimensional at each point. Its points are either boundary points, points where the unit sphere is one-dimensional but not closed, or an interior points, where the unit sphere is a circle, a simple closed graph without boundary.

The just verified combinatorial Gauss-Bonnet is entirely graph theoretical. Our curvature definition $K$ was motivated from the notion of Jacobi fields in the classical case given by second derivatives. While ”smoothness” requirements” are necessary for the more sophisticated Gauss-Bonnet formula, the just mentioned metric Gauss-Bonnet holds for any polyhedron with triangular faces. For example, every finite triangularization of a two-dimensional compact manifold works. Let us explain a bit more, why the curvature

is ”differential geometric”: the Gauss-Jacobi equations $f^{\prime\prime}=-Kf$ in differential geometry require the second differences of a Jacobi field $f$. [1]. Our starting point had been to extend Jacobi fields to the discrete for numerical purposes: for a discretized Jacobi field with smallest space step $1$, we have $f^{\prime\prime}(0)=f(2)-2f(1)+f(0)=f(2)-2f(1)$. The Jacobi equations suggest to call this $-K$. Since $f(k)$ is the variation of the geodesic when changing the angle, we can integrate over the circle and we get the length $|S_{k}|$ of the circle of radius $k$. Therefore $K$ is a multiple of $2|S_{1}|-|S_{2}|$ and there is no reason to normalize this in the discrete.
The ”first order curvature” $K_{1}=6-|S_{1}|$ on the other hand only requires first order differences. The curvature $K$ has some advantages over the curvature $K_{1}$:

• •

  the curvature formula for the boundary and in the interior is the same, while for the curvature $K_{1}$, one has to distinguish boundary and interior.
  

• •

  there is no reference to a flat background structure for $K$, while $K_{1}$ refers to the flat situation with via integers $6$ or $3$.
  

• •

  it can be generalized to more general situations, where the distance in the graph can vary and where we have no natural flatness as a reference. We can look for example for distance functions which minimize the total curvature.
  

• •

  it is more closely rooted to differential geometry of manifolds and classical notions like Jacobi fields, a notion which is of ”second order” too.
  

• •

  it can be adapted to higher dimension, when defining scalar curvature for graphs and where no natural ”flat triangulated ambient reference graph” exists.
  

To summarize, we think that while $K_{1}$ is ”metric”, $K$ has a more ”differential geometric” flavor. Similarly as many metric results extend to the differential geometric setup, things are more restricted also in the discrete, if higher order difference notions are used. The limitations of the results are related to similar limitations we know in the continuum.

We can combine the two results: for the Puiseux curvature with radius $2$ defined by

we get the following corollary:

Note that unlike the combinatorial curvature formula $\sum_{g\in G}K_{1}(g)=6\chi(G)$, the $K_{2}$ formula is only obvious modulo the main result for ”smooth domains” proved here. If we wanted to establish Gauss-Bonnet type results for curvatures like $K_{3}=3S_{1}-S_{3}$, the restrictions on discrete domains would be even more severe.