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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.

Quadratic forms (i.e. homogeneous polynomials of degree two) are fundamental mathematical objects. For the ancient Greeks, quadratic forms manifested in the geometry of conic sections, and in Pythagoras’ theorem. Riemann recognized the importance of studying abstract smooth manifolds equipped with a field of infinitesimal quadratic forms (i.e. a Riemannian metric), giving rise to the theory of Riemannian manifolds. In contrast to more general norms, an inner product on a vector space enjoys a big group of symmetries; thus infinitesimal Riemannian geometry inherits all the richness of the representation theory of orthogonal groups, which organizes the various curvature tensors and Weitzenbock formulae. It is natural that quadratic forms should come up in so many distinct ways in differential geometry: one uses calculus to approximate a smooth object near some point by a linear object, and the “difference” is a second-order term, which can often be interpreted as a quadratic form. For example:

  1. If M is a Riemannian manifold, at any point p one can choose an orthonormal frame for T_p M, and exponentiate to obtain geodesic normal co-ordinates. In such local co-ordinates, the metric tensor g_{ij} satisfies g_{ij}(p)=\delta_{ij} and \partial_kg_{ij}(p) = 0. The second order derivatives can be expressed in terms of the Riemann curvature tensor at p.
  2. If S is an immersed submanifold of Euclidean space, at every point p \in S there is a unique linear subspace that is tangent to S at p. The second order difference between these two spaces is measured by the second fundamental form of S, a quadratic form (with coefficients in the normal bundle) whose eigenvectors are the directions of (extrinsic) principal curvature. If S has codimension one, the second fundamental form is easily described in terms of the Gauss map g: S \to S^{n-1} taking each point on S to the unique unit normal to S at that point, and using the flatness of the ambient Euclidean space to identify the normal spheres at different points with “the” standard sphere. The second fundamental form is then defined by the formula II(v,w) = \langle dg(v),w \rangle. For higher codimension, one considers Gauss maps with values in an appropriate Grassmannian.
  3. If f is a smooth function on a manifold M, a critical point p of f is a point at which df=0 (i.e. at which all the partial derivatives of f in some local coordinates vanish). At such a point, one defines the Hessian Hf, which is a quadratic form on T_pM, determined by the second partial derivatives of f at such a point. If \nabla is a Levi-Civita connection on T^*M (determined by an Riemannian metric on M compatible with the smooth structure) then Hf = \nabla df. The condition that the Levi-Civita connection is torsion-free translates into the fact that the antisymmetric part of \nabla \theta is equal to d\theta for any 1-form \theta; in this context, this means that the antisymmetric part of the Hessian vanishes — i.e. that it is symmetric (and therefore a quadratic form). If \nabla' is a different connection, then \nabla' df = \nabla df + \alpha \wedge df for some 1-form \alpha, and therefore their values at p agree, and Hf is well-defined, independent of a choice of metric.

By contrast, cubic forms are less often encountered, either in geometry or in other parts of mathematics; their appearance is often indicative of unusual richness. For example: Lie groups arise as the subgroups of automorphisms of vector spaces preserving certain structure. Orthogonal and symplectic groups are those that preserve certain (symmetric or alternating) quadratic forms. The exceptional Lie group G_2 is the group of automorphisms of \mathbb{R}^7 that preserves a generic (i.e. nondegenerate) alternating 3-form. One expects to encounter cubic forms most often in flavors of geometry in which the local transformation pseudogroups are bigger than the orthogonal group.

One example is that of 1-dimensional complex projective geometry. If U is a domain in the Riemann sphere, one can think of U as a geometric space in at least two natural ways: by considering the local pseudogroup of all holomorphic self-maps between open subsets of the Riemann sphere, restricted to U (i.e. all holomorphic functions), or by considering only those holomorphic maps that extend to the entire Riemann sphere (i.e. the projective transformations: z \to \frac {az+b} {cz+d}). The difference between these two geometric structures is measured by a third-order term, called the Schwarzian derivative. If U is homeomorphic to a disk, then we can think of U as the image of the round unit disk D under a uniformizing map f. At every point p \in D there is a unique projective transformation f_p that osculates to f to second order at p (i.e. has the same value, first derivative, and second derivative as f at the point p); the (scaled) third derivative is the Schwarzian of f at p. In local co-ordinates, Sf = f'''/f' - \frac {3} {2} \left( f''/f'\right)^2. Actually, although the Schwarzian is sensitive to third-order information, it should really be thought of as a quadratic form on the (one-dimensional) complex tangent space to p.

Real projective geometry gives rise to similar invariants. Consider an immersed curve in the (real projective) plane. At every point, there is a unique osculating conic, that agrees with the immersed curve to second order. The projective curvature (really a cubic form) measures the third order deviation between these two immersed submanifolds at this point. See e.g. the book by Ovsienko and Tabachnikov for more details.

Another example is the so-called symplectic curvature. Let X be a flat symplectic space; this could be ordinary Euclidean space \mathbb{R}^{2n} with its standard symplectic form, or a quotient of such a space by a discrete group of translations. A linear subspace \pi of \mathbb{R}^{2n} through the origin is a Lagrangian subspace if it has (maximal) dimension n, and the restriction of the symplectic form to \pi is identically zero. A smooth submanifold L of dimension n is Lagrangian if its tangent space at every point is a Lagrangian submanifold. A Lagrangian submanifold of a flat symplectic space inherits a natural cubic form on the tangent space at every point, which can be defined in any of the following equivalent ways:

  1. If W is a symplectic manifold and L is a Lagrangian submanifold, then near any point p one can find a neighborhood U and choose symplectic coordinates so that U is symplectomorphic to a neighborhood of some point in T^*L. Moreover, every other Lagrangian submanifold L' sufficiently close (in C^1) to L can be taken in some possibly smaller neighborhood to be of the form df, where f is a smooth function on L (well-defined up to a constant), thought of as a section of T^*L. In the context above, choose local symplectic coordinates (by a linear symplectic transformation) for which the flat space looks locally like T^*\pi and L looks locally like df. The condition that \pi and L are tangent at the origin means that the 2-jet of f vanishes. The first nonvanishing term are the third partial derivatives of f, which can be thought of as the coefficients of a (symmetric) cubic form on \pi.
  2. If we choose a Euclidean metric on X compatible with the flat symplectic structure, the second fundamental form of L at some point is a quadratic form on \pi with coefficients in the normal bundle to \pi. The symplectic form identifies the normal \pi^\perp to \pi with the dual \pi^*, so by contracting indices, one obtains a cubic form on \pi. This form does not depend on the choice of Euclidean metric, since a different metric skews the normal bundle \pi^\perp replacing it with \pi^\perp + \alpha\pi. But since \pi is Lagrangian, the identification of this normal bundle with \pi^* is insensitive to the skewed term, and therefore independent of the choices.
  3. The space of all Lagrangian subspaces \Lambda of \mathbb{R}^{2n} is a symmetric space, homeomorphic to U(n)/O(n), sometimes called the Shilov boundary of the Siegel upper half-space. If \pi \in \Lambda and \pi'_0 is a tangent vector to \pi in \Lambda, then one obtains a symmetric quadratic form on \pi in the following way. If \sigma is a transverse Lagrangian to \pi, and \pi_t is a 1-parameter family of Lagrangians starting at \pi, then for small  t the Lagrangians \pi_t and \sigma are transverse, and span \mathbb{R}^{2n}. For any v \in \mathbb{R}^{2n} there is a unique decomposition v = v(\pi_t) + v(\sigma). Define q_t(v,w) = \omega(v(\pi_t),w(\sigma)). Then q'_0 is a symmetric bilinear form that vanishes on \sigma, and therefore descends to a form on \pi that depends only on \pi'_0. A Lagrangian submanifold L maps to \Lambda by the Gauss map g. One obtains a cubic form on \pi associated to L as follows: if u,v,w \in \pi then dg(u) is a tangent vector to \pi in \Lambda, and therefore determines a quadratic form on \pi; this form is then evaluated on the vectors v,w.

One application of symplectic curvature is to homological mirror symmetry, where the symplectic curvature associated to a Lagrangian family of Calabi-Yau 3-folds Y in H^3(Y) determines the so-called “Yukawa 3-differential”, whose expression in a certain local coordinate gives the generating function for the number of rational curves of degree d in a generic quintic hypersurface in \mathbb{CP}^4. This geometric picture is described explicitly in the work of Givental (e.g. here). In another more recent paper, Givental shows how the topological recursion relations, the string equation and the dilaton equation in Gromov-Witten theory can be reformulated in terms of the geometry of a certain Lagrangian cone in a formal loop space (the geometric property of this cone is that it is overruled — i.e. each tangent space L is tangent to the cone exactly along zL, where z is a formal variable). This geometric condition translates into properties of the symplectic curvature of the Lagrangian cone, from which one can read off the “gravitational descendents” in the theory (let me add that this subject is quite far from my area of expertise, and that I come to this material as an interested outsider).

Cubic forms occur naturally in other “special” geometric contexts, e.g. holomorphic symplectic geometry (Rozansky-Witten invariants), affine differential geometry (related to the discussion of the Schwarzian above), etc. Each of these contexts is the start of a long story, which is best kept for another post.

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