Characteristic classes of foliations

I recently learned from Jim Carlson’s blog of the passing of Harsh Pittie on January 16 this year. I never met Harsh, but he is very familiar to me through his work, in particular for his classic book Characteristic classes of foliations. I first encountered this book as a graduate student, late last millenium. The book has several features which make it very attractive. First of all, it is short — only 105 pages. My strong instinct is always to read books cover-to-cover from start to finish, and the motivated reader can go through Harsh’s book in its entirety in maybe 4 hours. Second of all, the exposition is quite lovely: an excellent balance is struck between providing enough details to meet the demands of rigor, and sustaining the arc of an idea so that the reader can get the point. Third of all, the book achieves a difficult synthesis of two “opposing” points of view, namely the (differential-)geometric and the algebraic. In fact, this achievement is noted and praised in the MathSciNet review of the book (linked above) by Dmitri Fuks.

Hyperbolic Geometry Notes #2 – Triangles and Gauss Bonnet

In this post, I will cover triangles and area in spaces of constant (nonzero) curvature. We are focused on hyperbolic space, but we will talk about spheres and the Gauss-Bonnet theorem.

1. Triangles in Hyperbolic Space

Suppose we are given 3 points in hyperbolic space . A triangle with these points as vertices is a set of three geodesic segments with these three points as endpoints. The fact that there is a unique triangle requires a (brief) proof. Consider the hyperboloid model: three points on the hyperboloid determine a unique 3-dimensional real subspace of which contains these three points plus the origin. Intersecting this subspace with the hyperboloid gives a copy of , so we only have to check there is a unique triangle in . For this, consider the Klein model: triangles are euclidean triangles, so there is only one with a given three vertices.

In hyperbolic space, it is still true that knowing enough side lengths and/or angles of a triangles determines it. For example, knowing two side lengths and the angle between them determines the triangle. Similarly, knowing all the angles determines it. However, not every set of angles can be realized (in euclidean space, for example, the angles must add to ), and the inequalities which must be satisfied are more complicated for hyperbolic space.

2. Ideal Triangles and Area Theorems

We can think about moving one (or more) of the points of a hyperbolic triangle off to infinity (the boundary of the disk). An ideal triangle is one with all three “vertices” (the vertices do not exist in hyperbolic space) on the boundary. Using a conformal map of the disk (which is an isometry of hyperbolic space), we can move any three points on the boundary to any other three points, so up to isometry, there is only one ideal triangle. We have fixed our metric, so we can find the area of this triangle. The logically consistent way to find this is with an integral since we will use this fact in our proof sketch of Gauss-Bonnet, but as a remark, suppose we know Gauss-Bonnet. Imagine a triangle very close to ideal. The curvature is , and the euler characteristic is . The sum of the exterior angles is just slightly under , so using Gauss-Bonnet, the area is very close to , and goes to as we push the vertices off to infinity.

One note is that suppose we know what the geodesics are, and we know what the area of an ideal triangle is (suppose we just defined it to be without knowing the curvature). Then by pasting together ideal triangles, as we will see, we could find the area of any triangle. That is, really the key to understanding area is knowing the area of an ideal triangle.

As mentioned above, there is a single triangle, up to isometry, with given angles, so denote the triangle with angles by .

2.1. Area

Knowing the area of an ideal triangle allows us to calculate the area of any triangle. In fact:

Theorem 1 (Gauss)

This geometric proof relies on the fact that the angles in the Poincare model are the euclidean angles in the model. Consider the generic picture:

We have extended the sides of and drawn the ideal triangle containing these geodesics. Since the angles are what they look like, we know that the area of is the area of the ideal triangle (), minus the sum of the areas of the smaller triangles with two points at infinity:

Thus it suffices to show that .

For this fact, we need another picture:

Define . The picture shows that the area of the left triangle (with two vertices at infinity and one near the origin) plus the area of the right triangle is the area of the top triangle plus the area of the (ideal) bottom triangle:

We also know some boundary conditions on : we know (this is a degenerate triangle) and (this is an ideal triangle). We therefore conclude that

Similarly,

And we can find by observing that

Similarly, if we know , then

And by subtracting , we find that . By induction, then, if is a dyadic rational times . This is a dense set, so we know for all by continuity. This proves the theorem.

3. Triangles On Spheres

We can find a similar formula for triangles on spheres. A lune is a wedge of a sphere:

A lune.

Since the area of a lune is proportional to the angle at the peak, and the lune with angle has area , the lune with angle has area . Now consider the following picture:

Notice that each corner of the triangle gives us two lunes (the lunes for are shown) and that there is an identical triangle on the rear of the sphere. If we add up the area of all 6 lunes associated with the corners, we get the total area of the sphere, plus twice the area of both triangles since we have triple-counted them. In other words:

Solving,

4. Gauss-Bonnet

If we encouter a triangle of constant curvature , then we can scale the problem to one of the two formulas we just computed, so

This formula allows us to give a slightly handwavy, but accurate, proof of the Gauss-Bonnet theorem, which relates topological information (Euler characteristic) to geometric information (area and curvature). The proof will precede the statement, since this is really a discussion.

Suppose we have any closed Riemannian manifold (surface) . The surface need not have constant curvature. Suppose for the time being it has no boundary. Triangulate it with very small triangles such that and . Then since the deviation between the curvature and the curvature at the midpoint is times the distance from the midpoint,

For each triangle , we can form a comparison triangle with the same edge lengths and constant curvature . Using the formula from the beginning of this section, we can rewrite the right hand side of the formula above, so

Now since the curvature deviates by times the distance from the midpoint, the angles in deviate from those in just slightly:

So we have

Therefore, summing over all triangles,

The right hand side is just the total angle sum. Since the angle sum around each vertex in the triangulation is ,

Where is the number of vertices, and is the number of triangles. The number of edges, , can be calculated from the number of triangles, since there are edges for each triangle, and they are each double counted, so . Rewriting the equation,

Taking the mesh size to zero, we get the Gauss-Bonnet theorem .

4.1. Variants of Gauss-Bonnet

• If is compact with totally geodesic boundary, then the formula still holds, which can be shown by doubling the surface, applying the theorem to the doubled surface, and finding that euler characteristic also doubles.
• If has geodesic boundary with corners, then

  

  Where the turning angle is the angle you would turn tracing the shape from the outside. That is, it is , where is the interior angle.

• Most generally, if has smooth boundary with corners, then we can approximate the boundary with totally geodesic segments; taking the length of these segments to zero gives us geodesic curvature ():

  

4.2. Examples

• The Euler characteristic of the round disk in the plane is , and the disk has zero curvature, so . The geodesic curvature is constant, and the circumference is , so , so .
• A polygon in the plane has no curvature nor geodesic curvature, so .

The Gauss-Bonnet theorem constrains the geometry in any space with nonzero curvature. This the “reason” similarities which don’t preserve length and/or area exist in euclidean space; it has curvature zero.

Causal geometry

On page 10 of Besse’s famous book on Einstein manifolds one finds the following quote:

It would seem that Riemannian and Lorentzian geometry have much in common: canonical connections, geodesics, curvature tensor, etc. . . . But in fact this common part is only a common disposition at the onset: one soon enters different realms.

I will not dispute this. But it is not clear to me whether this divergence is a necessary consequence of the nature of the objects of study (in either case), or an artefact of the schism between mathematics and physics during much of the 20th century. In any case, in this blog post I have the narrow aim of describing some points of contact between Lorentzian (and more generally, causal) geometry and other geometries (hyperbolic, symplectic), which plays a significant role in some of my research.

The first point of contact is the well-known duality between geodesics in the hyperbolic plane and points in the (projectivized) “anti de-Sitter plane”. Let denote a -dimensional vector space equipped with a quadratic form

If we think of the set of rays through the origin as a copy of the real projective plane , the hyperbolic plane is the set of projective classes of vectors with , the (projectivized) anti de-Sitter plane is the set of projective classes of vectors with , and their common boundary is the set of projective classes of (nonzero) vectors with . Topologically, the hyperbolic plane is an open disk, the anti de-Sitter plane is an open Möbius band, and their boundary is the “ideal circle” (note: what people usually call the anti de-Sitter plane is actually the annulus double-covering this Möbius band; this is like the distinction between spherical geometry and elliptic geometry). Geometrically, the hyperbolic plane is a complete Riemannian surface of constant curvature , whereas the anti de-Sitter plane is a complete Lorentzian surface of constant curvature .

In this projective model, a hyperbolic geodesic is an open straight line segment which is compactified by adding an unordered pair of points in the ideal circle. The straight lines in the anti de-Sitter plane tangent to the ideal circle at these two points intersect at a point . Moreover, the set of geodesics in the hyperbolic plane passing through a point are dual to the set of points in the anti de-Sitter plane that lie on a line which does not intersect the ideal circle. In the figure, three concurrent hyperbolic geodesics are dual to three colinear anti de-Sitter points.

The anti de-Sitter geometry has a natural causal structure. There is a cone field whose extremal vectors at every point are tangent to the straight lines through that are also tangent to the ideal circle. A smooth curve is timelike if its tangent at every point is supported by this cone field, and spacelike if its tangent is everywhere not supported by the cone field. A timelike curve corresponds to a family of hyperbolic geodesics which locally intersect each other; a spacelike curve corresponds to a family of disjoint hyperbolic geodesics that foliate some region.

One can distinguish (locally) between future and past along a timelike trajectory, by (arbitrarily) identifying the “future” direction with a curve which winds positively around the ideal circle. The fact that one can distinguish in a consistent way between the positive and negative direction is equivalent to the existence of a nonzero section of timelike vectors. On the other hand, there does not exist a nonzero section of spacelike vectors, so one cannot distinguish in a consistent way between left and right (this is a manifestation of the non-orientability of the Möbius band).

The duality between the hyperbolic plane and the anti de-Sitter plane is a manifestation of the fact that (at least at the level of Lie algebras) they have the same (infinitesimal) symmetries. Let denote the group of real matrices which preserve ; i.e. matrices for which for all vectors . This contains a subgroup of index which preserves the “positive sheet” of the hyperboloid , and acts on it in an orientation-preserving way. The hyperbolic plane is the homogeneous space for this group whose point stabilizers are a copy of (which acts as an elliptic “rotation” of the tangent space to their common fixed point). The anti de-Sitter plane is the homogeneous space for this group whose point stabilizers are a copy of (which acts as a hyperbolic “translation” of the geodesic in hyperbolic space dual to the given point in anti de-Sitter space). The ideal circle is the homogeneous space whose point stabilizers are a copy of the affine group of the line. The hyperbolic plane admits a natural Riemannian metric, and the anti de-Sitter plane a Lorentz metric, which are invariant under these group actions. The causal structure on the anti de-Sitter plane limits to a causal structure on the ideal circle.

Now consider the -dimensional vector space and the quadratic form . The (-dimensional) sheets and both admit homogeneous Lorentz metrics whose point stabilizers are copies of and (which are isomorphic but sit in in different ways). These -manifolds are compactified by adding the projectivization of the cone . Topologically, this is a Clifford torus in dividing this space into two open solid tori which can be thought of as two Lorentz -manifolds. The causal structure on the pair of Lorentz manifolds limits to a pair of complementary causal structures on the Clifford torus. (edited 12/10)

Let’s go one dimension higher, to the -dimensional vector space and the quadratic form . Now only the sheet is a Lorentz manifold, whose point stabilizers are copies of , with an associated causal structure. The projectivized cone is a non-orientable twisted bundle over the circle, and it inherits a causal structure in which the sphere factors are spacelike, and the circle direction is timelike. This ideal boundary can be thought of in quite a different way, because of the exceptional isomorphism at the level of (real) Lie algebras , where denotes the Lie algebra of the symplectic group in dimension . In this manifestation, the ideal boundary is usually denoted , and can be thought of as the space of Lagrangian planes in with its usual symplectic form. One way to see this is as follows. The wedge product is a symmetric bilinear form on with values in . The associated quadratic form vanishes precisely on the “pure” -forms — i.e. those associated to planes. The condition that the wedge of a given -form with the symplectic form vanishes imposes a further linear condition. So the space of Lagrangian -planes is a quadric in , and one may verify that the signature of the underlying quadratic form is . The causal structure manifests in symplectic geometry in the following way. A choice of a Lagrangian plane lets us identify symplectic with the cotangent bundle . To each symmetric homogeneous quadratic form on (thought of as a smooth function) is associated a linear Lagrangian subspace of , namely the (linear) section . Every Lagrangian subspace transverse to the fiber over is of this form, so this gives a parameterization of an open, dense subset of containing the point . The set of positive definite quadratic forms is tangent to an open cone in ; the field of such cones as varies defines a causal structure on which agrees with the causal structure defined above.

These examples can be generalized to higher dimension, via the orthogonal groups or the symplectic groups . As well as two other infinite families (which I will not discuss) there is a beautiful “sporadic” example, connected to what Freudenthal called octonion symplectic geometry associated to the noncompact real form of the exceptional Lie group, where the ideal boundary has an invariant causal structure whose timelike curves wind around the factor; see e.g. Clerc-Neeb for a more thorough discussion of the theory of Shilov boundaries from the causal geometry point of view, or see here or here for a discussion of the relationship between the octonions and the exceptional Lie groups.

The causal structure on these ideal boundaries gives rise to certain natural -cocycles on their groups of automorphisms. Note in each case that the ideal boundary has the topological structure of a bundle over with spacelike fibers. Thus each closed timelike curve has a well-defined winding number, which is just the number of times it intersects any one of these spacelike slices. Let be an ideal boundary as above, and let denote the cyclic cover dual to a spacelike slice. If is a point in , we let denote the image of under the th power of the generator of the deck group of the covering. If is a homeomorphism of preserving the causal structure, we can lift to a homeomorphism of . For any such lift, define the rotation number of as follows: for any point and any integer , let be the the smallest integer for which there is a causal curve from to to , and then define . This function is a quasimorphism on the group of causal automorphisms of , with defect equal to the least integer such that any two points in are contained in a closed causal loop with winding number . In the case of the symplectic group with causal boundary , the defect is , and the rotation number is (sometimes) called the symplectic rotation number; it is a quasimorphism on the universal central extension of , whose coboundary descends to the Maslov class (an element of -dimensional bounded cohomology) on the symplectic group.

Causal structures in groups of symplectomorphisms or contactomorphisms are intensely studied; see for instance this paper by Eliashberg-Polterovich.