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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 \mathbb{R}^{2,1} denote a 3-dimensional vector space equipped with a quadratic form

q(x,y,z) = x^2 + y^2 - z^2

If we think of the set of rays through the origin as a copy of the real projective plane \mathbb{RP}^2, the hyperbolic plane is the set of projective classes of vectors v with q(v)<0, the (projectivized) anti de-Sitter plane is the set of projective classes of vectors v with q(v)>0, and their common boundary is the set of projective classes of (nonzero) vectors v with q(v)=0. 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 -1, whereas the anti de-Sitter plane is a complete Lorentzian surface of constant curvature -1.

In this projective model, a hyperbolic geodesic \gamma 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 p_\gamma. Moreover, the set of geodesics \gamma in the hyperbolic plane passing through a point q are dual to the set of points p_\gamma 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 p are tangent to the straight lines through p 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 O(2,1) denote the group of real 3\times 3 matrices which preserve q; i.e. matrices A for which q(A(v)) = q(v) for all vectors v. This contains a subgroup SO^+(2,1) of index 4 which preserves the “positive sheet” of the hyperboloid q=-1, 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 SO(2) (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 SO^+(1,1) (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 4-dimensional vector space \mathbb{R}^{2,2} and the quadratic form q(v) = x^2 + y^2 - z^2 - w^2. The (3-dimensional) sheets q=1 and q=-1 both admit homogeneous Lorentz metrics whose point stabilizers are copies of SO^+(1,2) and SO^+(2,1) (which are isomorphic but sit in SO(2,2) in different ways). These 3-manifolds are compactified by adding the projectivization of the cone q=0. Topologically, this is a Clifford torus in \mathbb{RP}^3 dividing this space into two open solid tori which can be thought of as two Lorentz 3-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 5-dimensional vector space \mathbb{R}^{2,3} and the quadratic form q(v) = x^2 + y^2 - u^2 - z^2 - w^2. Now only the sheet q=1 is a Lorentz manifold, whose point stabilizers are copies of SO^+(1,3), with an associated causal structure. The projectivized cone q=0 is a non-orientable twisted S^2 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 so(2,3)= sp(4), where sp(4) denotes the Lie algebra of the symplectic group in dimension 4. In this manifestation, the ideal boundary is usually denoted \mathcal{L}_2, and can be thought of as the space of Lagrangian planes in \mathbb{R}^4 with its usual symplectic form. One way to see this is as follows. The wedge product is a symmetric bilinear form on \Lambda^2 \mathbb{R}^4 with values in \Lambda^4 \mathbb{R}^4 = \mathbb{R}. The associated quadratic form vanishes precisely on the “pure” 2-forms — i.e. those associated to planes. The condition that the wedge of a given 2-form with the symplectic form vanishes imposes a further linear condition. So the space of Lagrangian 2-planes is a quadric in \mathbb{RP}^4, and one may verify that the signature of the underlying quadratic form is (2,3). The causal structure manifests in symplectic geometry in the following way. A choice of a Lagrangian plane \pi lets us identify symplectic \mathbb{R}^4 with the cotangent bundle T^*\pi. To each symmetric homogeneous quadratic form q on \pi (thought of as a smooth function) is associated a linear Lagrangian subspace of T^*\pi, namely the (linear) section dq. Every Lagrangian subspace transverse to the fiber over 0 is of this form, so this gives a parameterization of an open, dense subset of \mathcal{L}_2 containing the point \pi. The set of positive definite quadratic forms is tangent to an open cone in T_\pi \mathcal{L}_2; the field of such cones as \pi varies defines a causal structure on \mathcal{L}_2 which agrees with the causal structure defined above.

These examples can be generalized to higher dimension, via the orthogonal groups SO(n,2) or the symplectic groups Sp(2n,\mathbb{R}). 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 E_7(-25) of the exceptional Lie group, where the ideal boundary S^1\times E_6/F_4 has an invariant causal structure whose timelike curves wind around the S^1 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 2-cocycles on their groups of automorphisms. Note in each case that the ideal boundary has the topological structure of a bundle over S^1 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 C be an ideal boundary as above, and let \tilde{C} denote the cyclic cover dual to a spacelike slice. If p is a point in \tilde{C}, we let p+n denote the image of p under the nth power of the generator of the deck group of the covering. If g is a homeomorphism of C preserving the causal structure, we can lift g to a homeomorphism \tilde{g} of \tilde{C}. For any such lift, define the rotation number of \tilde{g} as follows: for any point p \in \tilde{C} and any integer n, let r_n be the the smallest integer for which there is a causal curve from p to \tilde{g}(p) to p+r_n, and then define rot(\tilde{g}) = \lim_{n \to \infty} r_n/n. This function is a quasimorphism on the group of causal automorphisms of \tilde{C}, with defect equal to the least integer n such that any two points p,q in C are contained in a closed causal loop with winding number n. In the case of the symplectic group Sp(2n,\mathbb{R}) with causal boundary \mathcal{L}_n, the defect is n, and the rotation number is (sometimes) called the symplectic rotation number; it is a quasimorphism on the universal central extension of Sp(2n,\mathbb{R}), whose coboundary descends to the Maslov class (an element of 2-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.

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