Simplicial curvature & simplicial Gauss-Bonnet

December 24, 2008 — rip

Introduction and text

The last section of Bloch’s chapter 3 (simplicial surfaces) is a long (and to my mind at this time, uninteresting) proof of the 2D Brouwer fixed point theorem: any continuous map from the disk to the disk has a fixed point. Bloch also proves a corollary, the no-retraction theorem, that there is no continuous map r from the disk to the circle such that r(x) = x for all x on the circle.

That one sounds interesting. We’ve seen in before, with the commentary that you can’t map the surface of a drum onto its rim without tearing it. I still don’t see it that way. But it is rather shocking that the map r cannot preserve all the points on the rim.

Anyway, we’re not going to fight with those. For me, the climax of chapter 3 is the simplicial Gauss-Bonnet theorem. It shows that there is a definition of curvature for simplicial surfaces (in fact, for polyhedra in general) such that the total curvature of a surface is equal to 2\ \pi times its Euler characteristic \chi\ .

(A simplicial surface is a polyhedron all of whose faces are triangles. I expect we’ll see this again in another post.)

That the total Gaussian curvature of a surface is equal to 2\ \pi\ \chi is called the Gauss-Bonnet theorem. It is a reasonable culmination of a first course in differential geometry. The simplicial version means that we have a definition of curvature for simplicial surfaces and polyhedra which gives us a form of the Gauss-Bonnet theorem. That says it’s a reasonable definition of curvature.

So what is this marvelous definition of simplicial curvature? It’s also called the angle defect, and goes back to Descartes.

First off, it turns out that all the curvature of a polyhedron (or simplicial surface) is concentrated at the vertices. There is no contribution from the edges.

(Either recall or trust me that the Gaussian curvature of a cylinder is zero…. The simplicial curvature of an infinitely long polygonal cylinder ought to be zero. That is, if instead of a circular cross-section, our cylinder has a polygonal cross-section, we would like the total curvature to still be zero. I take it to be infinitely long so that it has no vertices, only edges and faces. Such a thing is not a compact surface, but it may help justify the idea that edges do not contribute to simplicial curvature.)

Imagine some polyhedron. There are at least two ways to describe the simplcial curvature. One is to look at each vertex. For a fixed vertex v, consider the angle at v for each of the faces \eta which touch v, add up those angles, and subtract the sum from 2\ \pi\ . (That’s why it’s also called the angle defect: it measures how far from 2\ \pi\ , i.e. how far from planar, the polygons are when they’re touching. The pictures that follow may help.)

Note that we are looking at one angle from each polygon that touches the vertex v; we are not measuring an angle in space at the vertex. Anyway, add up all the angles at each vertex, and subtract that sum from 2\ \pi\ , and call it the angle defect at v.

Then add up all the angle defects: that sum is the total simplicial curvature, and it’s equal to 2\ \pi\ \chi\ for that surface.

The other way to do it is to count all the angles first. Add up all the angles in all the faces. Subtract that sum from 2\ \pi\ V\ , where V is the number of vertices. (Each angle in a polygon is associated with one vertex.)

I believe I could write that compactly as:

Let K be a polyhedron, and let v \in K be a vertex. If \sigma \in K is a face (polygon) containing v, let \angle(v,\ \sigma) denote the angle at v in \sigma\ . The curvature of K at v (or the angle defect at v) is defined to be the number d(v) given by

d(v) = 2\ \pi\ - \sum_{\eta}\angle(v,\ \eta)

where the \eta are the faces of K containing v.

The total simplicial curvature of the polyhedron is the sum of the angle defects,

\sum_{v} d(v)\ .

Let’s look at some examples.

regular tetrahedron

The simplest possible case is a regular tetrahedron: put 4 equilateral triangles together.

reg-tetra

There are 4 vertices and every vertex is the same. At each vertex, 3 equilateral triangles meet, so the sum of the angles at one vertex is

3\ \frac{\pi}{3} = \pi

so the angle defect at one vertex is

2\ \pi - \pi = \pi

and there are 4 vertices, so the total simplicial curvature is 4\ \pi\ .

By direct computation, the Euler characteristic of a tetrahedron is \chi = v - e + f = 4 - 6 + 4 = 2\ .

Therefore, we do indeed have

2\ \pi\ \chi = 4 \pi = \text{total simplicial curvature}\ .

slighly irregular tetrahedron

What about an irregular tetrahedron? Suppose we put the vertices at the origin and on each axis at a distance of 1. There are 3 faces on each of the coordinate planes: these faces are 1 - 1 - \sqrt{2} right triangles. The 4th face is an equilateral triangle.

irreg-tetra

The 4th face does not touch the vertex at the origin (shown orange); we have 3 right angles there. The sum of the angles is 3\ \frac{\pi}{2}\ ,

so the angle defect is

2\ \pi - 3\ \frac{\pi}{2} = \frac{\pi}{2}\ .

(Oh, of course! Look at the picture. More importantly, look at the open space at the orange vertex. That open space has an angle of \frac{\pi}{2}\ . That’s where the 2 \pi comes from, and why we subtract from it.

At the other three vertices (shown yellow, blue, black; the black vertex is in 3 pieces), we have one equilateral face \left(60{}^{\circ} \right) and two 45{}^{\circ} angles from the other two faces. At each of these 3 faces, the sum of angles is

\frac{\pi}{3} + 2\ \frac{\pi}{4} = \frac{5\ \pi}{6}\ ,

so the angle defect is

2\ \pi - \frac{5\ \pi}{6} = \frac{7\ \pi}{6}\ .

If we look at either the yellow or blue vertices, the portion that is not faces is certainly larger than 180{}^{\circ} \ , and in fact we can see that it’s 210{}^{\circ} \ , which is, indeed, \frac{7\ \pi}{6}\ .

Now we add up all the angle defects

\frac{\pi}{2} + 3\ \frac{7\ \pi}{6} = 4\ \pi\ .

Once again, that’s 2\ \pi\ \chi = 2\ \pi\ 2 = 4\ \pi\ .

What about a cube?

cube

Looking at any of the vertices shown as blue dots, we should guess that the angle defect is \frac{\pi}{2}\ at each of the 8 vertices, so the total is, once again,

8\ \frac{\pi}{2} = 4\ \pi\ .

Alternatively, 3 rectangles meet at each vertex, so the sum of the angles is 3\ \frac{\pi}{2}

and the angle defect at each vertex is directly computed to be

2\ \pi - 3\ \frac{\pi}{2} = \frac{\pi}{2}\ .

OTOH, the Euler characteristic of a cube is

\chi = v - e + f = 8 - 12 + 6 = 2\ ,

so 2 \pi\ \chi = 4 \pi\ .

(I know perfectly well that the cube is homeomorphic to the sphere, and therefore has the same Euler characteristic, but come on: we compute the Euler characteristics of polyhedra and simplicial surfaces, and then define the Euler characteristic of the sphere from them. I will admit, however, that if don’t get 2 by direct computation (for polyhedra without holes), I know I made a mistake!)

hexagonal cylinder (prism)

How about a finite polygonal cylinder? We know it’s homeomorphic to a sphere (\chi = 2), so we know the total simplicial curvature should be 4 \pi\ , but let’s work it out.

Here’s a regular hexagonal cylinder, or prism. There are 12 vertices; each is touched by 2 rectangles \left(90{}^{\circ} \right) and a hexagon \left(120{}^{\circ} \right).

prism-6

(Oh, cut the hexagon into 6 equilateral triangles; each angle at a vertex of the hexagon is made up of two 60{}^{\circ} \ angles.)

At each vertex, then, the sum of the angles is

2\ \frac{\pi}{2} + \frac{2\ \pi}{3} = \frac{5\ \pi}{3}\ ,

so the angle defect is \frac{\pi}{3}

and then the sum of those (i.e. 12 times the individual defect) is the simplicial curvature,

12\ \frac{\pi}{3} = 4\ \pi\ .

torus

How about a torus? First off, let’s grab a triangulation from here.

triangulation-18

Just look at any interior vertex: there is no gap, no space where a face isn’t. The angle defect should be zero at every interior vertex.

But every vertex is interior. The ones that seem to be on the boundary are identified with those on another boundary.

Each of the angle defects is zero, the sum of the angle defects is zero, and we need

2\ \pi\ \chi = 0\ ,

But that triangulation has 18 faces (that’s the easy number), 27 edges, and 9 vertices. \chi = 9 - 27 + 18 = 0\ .

Good. We’ve finally seen a case that didn’t add up to 4\ \pi\ .

Gaussian curvature of the smooth torus

One last thing. The smooth torus, like the cylinder, has total gaussian curvature equal to zero. Unlike the cylinder, however, the torus does not have constant curvature equal to zero everywhere.

(When we take a finite cylinder and bend it to make a torus, we have to stretch the material that goes to the outside and compress the material that goes to the inside.)

In case you’re curious, if we parameterize a torus as

\{r \sin (u),(R+r \cos (u)) \cos(v),(R+r \cos (u)) \sin (v)\}

then I believe the Gaussian curvature (which I have not shwn you how to compute) is cos(u). (Really? Independent of the radii r and R? Yes! They show up in both the curvature and the element of area dA, and cancel out.) Integrated over u \in [0,\ 2\pi]\ , that will be zero. And zero integrated over v \in [0,\ 2\pi]\ will still be zero.

What did I actually get from that parameterization? (Take r = 1, R = 3.)
torus

The red curve (outer rim) is u = 0, so cos u = 1 and the curvature > 0.
The black curve (inner rim) is u = \pi\ , so cos u = -1 and the curvature < 0.
The green curve is u = \frac{\pi}{2}\ , so cos u = 0 = curvature.

So we have seen some examples of computing the curvature of simplicial surfaces and polyhedra.

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The Euler Characteristic: Teasers

November 24, 2008 — rip

These are things i came across when I first started looking at the Euler characteristic, in fact, when I was looking at triangulations in particular.

n-manifolds

The Euler characteristic \chi generalizes to dimensions other than 2, and there are at least three noteworthy theorems involving the Euler characteristic. I’m not going to say much about them, because they, like so much else, are still outside my comfort zone. I’ll just barely tell you what they are, and leave you to chase them down if they interest you.

As we’ve seen, the Euler characteristic of a polyhedron is given by

\chi = v - e + f\ ,

where v, e, f are the numbers of vertices, edges, and faces. Homeomorphic polyhedra have the same Euler characteristic, and that means we can define the Euler characteristic of a topological surface as the Euler characteristic of any polyhedron which is homeomorphic to it.

This alternating-sign sum of the numbers of 0-, 1-, and 2- simplices generalizes in the obvious way: for an n-simplex, we take the sum, with alternating signs, of the numbers of k-simplices, for k <= n. As for surfaces, so for n-manifolds: this is a topological invariant, and we want to define the Euler characteristic of an n-manifold as the Euler characteristic of any k-simplex homeomorphic to it.

But we’ve been told that topological, simplicial, and differential structures on manifolds do not always coincide for n > 3. We can define the Euler characteristic for k-simplices and for manifolds with simplicial structure, but I am going to look skeptically at the Euler characteristic of topological and differentiable manifolds until I learn more about the different structures.

BTW, we can go down a dimension, too: we can define the Euler characteristic of a graph as v – e.

Betti numbers

The Euler characteristic of an n-manifold has an extremely interesting equivalence: not only does the formula generalize from 2 to n, but it can also be generalized to a topological space X: we can write

\chi = \sum_{ k = 1}^{ N} (-1)^k \beta_k

where \beta_k is the kth Betti number of X, and we assume that eventually (i.e. for k large enough) \beta_k= 0\ . (Lee, “Introduction to Topological Manifolds”; or Rotman, “An Introduction to Algebraic Topology”.)

But hold on just a minute. The Euler characteristic was defined for things with simplicial structure, and the Betti numbers are defined (not by me!) for topological spaces. But we know that simplicial & topological do not coincide for higher dimensions than 3, so we probably have to be rather careful here.

When all else fails, look carefully at the book. What we actually have is two definitions, a theorem, and an extension of the definition. We define the Euler characteristic of a simplicial complex, and the Betti numbers of a topological space; the theorem says that for a finite simplicial complex, the Euler characteristic is equal to the alternating sum of the Betti numbers:

\chi = \sum_{ k = 1}^{ N} (-1)^k \beta_k

Having the theorem, we then extend the definition of the Euler characteristic to any topological space: we define the Euler characteristic as the alternating sum of the Betti numbers; the theorem assures us that the two definitions of the Euler characteristic coincide on simplicial complexes.

We’re still not out of the woods. Looking in Rotman, I am reminded that there are different homology theories. Do they all give the same Betti numbers? I have no idea. (Well, I suppose they do, but I’m ready for anything.) Gee, no wonder algebraic topology is still on my list of things to study.

As an aside, in Massey’s combined book, “A Basic Course in Algebraic Topology”, I find a note which says that until about 1930, mathematicians focused on the Betti numbers (and the torsion coefficients) of homology groups, rather than on the groups themselves.

Index of a surface

Next, on this detour, we have the Poincare-Hopf index theorem, which says that the Euler characteristic of a surface is equal to the index of the surface.

Hey what?

As the Betti numbers came out of left field, the study of vector fields on a surface is coming out of right field. If you’ve ever been told that you can’t comb the hair on a ball, you’ve encountered one of the key theorems of the subject.

There are two ways to look at the index theorem. In the special setting of dynamical systems (i.e. differential equations), we are looking at points where the vector field vanishes (critical points, which are equilibrium points, but not necessarily stable equilibria). More specifically, we are interested in the flow pattern in the vicinity of critical points.

In this context, we break flow patterns into sectors each of which is called parabolic, hyperbolic, or elliptic; and we have a recipe (due to Bendixson) for planar systems which says that the index of a critical point (wrt the vector field) is

I = 1 + (e-h)/2,

where e and h are the numbers of elliptic and hyperbolic sectors. (The parabolic sectors don’t contribute to the index.)

The index of the surface wrt the vector field is the sum of the indices of the critical points. The Poincare-Hopf theorem says that the

index of the surface (in fact, of an n-manifold) wrt the vector field is equal to the Euler characteristic of the manifold.

(I believe that Poincare proved it for surfaces and Hopf extended it to higher dimensions. I am not sure that the definition of sectors applies in higher dimensions; that is, I am not sure how the index is defined in higher dimensions.)

This imposes significant constraints on the possible vector fields.

That’s one way to do it. My references for this approach are Perko, “Differential Equations and Dynamical Systems” and Firby & Gardiner, “Surface Topology”.

I was hoping to at least construct illustrations of the three kinds of sectors, but I’m finding it difficult and very annoying to update my version 5 drawings to version 6 of Mathematica®. (Backwards compatibility? What’s that?)

From Firby & Gardiner, I extract the following description. A sector is parabolic if all paths lead to the critical point, or all paths lead away. A sector is elliptic if all paths begin and end at the critical point. A sector is hyperbolic if all paths sweep past the critical point.

But there is another way to define and compute the index of a critical point. In fact, we can compute the index of any point wrt a curve, but if it’s not a critical point, then its index is zero; therefore we can confine our attention to critical points. This approach starts with winding numbers. Fulton, “Algebraic Topology: A First Course”, is excellent for surfaces (with multiple chapters on winding numbers and vector fields), and even suggests how to generalize to higher dimensions. Also, Sieradski, “An Introduction to Topology and Homotopy”, has a few sections, rather than chapters, on winding numbers and vector fields.

I’m beginning to look forward to tackling Fulton.

Gauss-Bonnet

The Gauss-Bonnet theorem, as O’Neill states it (“Elementary Differential Geometry”) says that the gaussian curvature, integrated over all of a compact orientable surface, is equal to 2 \pi \chi\ , where \chi is the Euler characteristic of the surface.

Oh, but doesn’t that require differentiable structure?

Yes and no.

Bloch shows us the Gauss-Bonnet theorem for simplicial surfaces! It turns out we can associate curvature with the vertices of a simplicial surface, in such a way that the total curvature (the sum of the curvatures at each vertex) is equal to 2 \pi \chi\ .

I’ll talk about this when I discuss chapter 3 of Bloch. I finished the chapter eons ago, but I haven’t tallied it up yet.

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books for the Poincare Conjecture

February 4, 2008 — rip

 The following books have been added to the bibliography.

First I read O’Shea’s “Poincare”; it’s way past time it showed up in the bibliography. Then I bought Weeks and “Ricci” based on O’Shea’s “further reading”. One is high school – barely – and the other is high research. I’ve skimmed Weeks once and have settled down to read it again. “Ricci” is beyond me, as I expected.

While trying to find out how a surface could be given a geometry in which it had constant curvature, I ended up at the back of O’Neill; he’s an old favorite. While trying to sort out topological and differential structures, I took another look at “Instantons” – right up there with “Ricci Flow” – and then discovered why I had bought Bloch in the first place: I could read him. 

Bloch, Ethan D.; A First Course in Geometric Topology and Differential Geometry.
Birkhauser 1997; ISBN 0 8176 3840 7.
[topology, geometry;4 feb 2008]
upper division mathematics.
it is about 3 kinds of structure on surfaces: topological, simplicial, and differential. the restriction to surfaces makes it a quite readable introduction. if you’re like me, you’ve seen each of these structures, but never together.

Freed, Daniel S. and Uhlenbeck, Karen K.; Instantons and Four-Manifolds.
Springer, 1984. ISBN 0 387 96036 8.
[topology, geometry;4 feb 2008]
from the first paragraph of the preface: “This book is the outcome of a seminar…. to go through a proof of Simon Donaldson’s Theorem…. the nonsmoothability of certain topological four-manifolds…. by studying the solution space of … the Yang-Mills equations….”
in other words, this is a highly advanced book. but who could resist the title? certainly not me.

O’Neill, Barrett; Elementary Differential Geometry.
Academic Press, 1997 (2nd Ed); ISBN 0 12 526745 2.
[differential geometry;4 feb 2008]
the first edition revolutionized the teaching of the subject, by introducing differential forms to undergraduates. the prerequisites are multivariate calculus and linear algebra. the 2nd ed. has more material on intrinsic geometry, e.g. more material on the gauss-bonnet theorem. 
this is one of “the books”. it’s also one of the very small set of which i have deliberately bought a later edition primarily to thank the author for the first edition.

O’Shea, Donal; The Poincare’ Conjecture;
Walker Publishing Co., 2007.  OSBN 0 8027 1532 X
[history of math, topology, geometry;4 feb 2008]
“This book is about a single problem. Formulated by a brilliant French mathematician, Henri Poincare, over one hundred years ago….” a pleasant mix of biography and history, and it mentions some fascinating mathematics, but – quite appropriately – it barely scratches the surface. i want more, but this is a good introduction.

Morgan, John and Tian, Gang; Ricci Flow and the Poincare Conjecture.
American Mathematical Society, 2007; ISBN 0 8218 4328 4
[topology;4 feb 2008]
this is the proof of the poincare conjecture. i may never understand it, but i couldn’t resist it. 

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