We consider foliations of the whole three dimensional hyperbolic space H 3 by oriented geodesics. Let L be the space of all the oriented geodesics of H 3 , which is a four dimensional manifold carrying two canonical pseudo-Riemannian metrics of signature (2, 2). We characterize, in terms of these geometries of L, the subsets M in L that determine foliations of H 3 . We describe in a similar way some distinguished types of geodesic foliations of H 3 , regarding to which extent they are in some sense trivial in some directions: On the one hand, foliations whose leaves do not lie in a totally geodesic surface, not even at the infinitesimal level. On the other hand, those for which the forward and backward Gauss maps ϕ ± : M → H 3 (∞) are local diffeomorphisms. Besides, we prove that for this kind of foliations, ϕ ± are global diffeomorphisms onto their images.The subject of this article is within the framework of foliations by congruent submanifolds, and follows the spirit of the paper by Gluck and Warner where they understand the infinite dimensional manifold of all the great circle foliations of the three sphere.
Let H be the hyperbolic space of dimension n+1. A geodesic foliation of H is given by a smooth unit vector field on H all of whose integral curves are geodesics. Each geodesic foliation of H determines an n-dimensional submanifold M of the 2n-dimensional manifold L of all the oriented geodesics of H (up to orientation preserving reparametrizations). The space L has a canonical split semi-Riemannian metric induced by the Killing form of the isometry group of H. Using a split special Lagrangian calibration, we study the volume maximization problem for a certain class of geometrically distinguished geodesic foliations, whose corresponding submanifolds of L are space-like.Mathematics Subject Classification: 53C38, 53C12, 53C22, 53C50
Let G be a Lie group of even dimension and let (g, J) be a left invariant anti-Kähler structure on G. In this article we study anti-Kähler structures considering the distinguished cases where the complex structure J is abelian or bi-invariant. We find that if G admits a left invariant anti-Kähler structure (g, J) where J is abelian then the Lie algebra of G is unimodular and (G, g) is a flat pseudo-Riemannian manifold. For the second case, we see that for any left invariant metric g for which J is an anti-isometry we obtain that the triple (G, g, J) is an anti-Kähler manifold.Besides, given a left invariant anti-Hermitian structure on G we associate a covariant 3-tensor θ on its Lie algebra and prove that such structure is anti-Kähler if and only if θ is a skew-symmetric and pure tensor. From this tensor we classify the real 4-dimensional Lie algebras for which the corresponding Lie group has a left invariant anti-Kähler structure and study the moduli spaces of such structures (up to group isomorphisms that preserve the anti-Kähler structures).2010 Mathematics Subject Classification.
Let N be a pseudo-Riemannian manifold such that L 0 (N ), the space of all its oriented null geodesics, is a manifold. B. Khesin and S. Tabachnikov introduce a canonical contact structure on L 0 (N ) (generalizing the definition given by R. Low in the Lorentz case), and study it for the pseudo-Euclidean space. We continue in that direction for other spaces.Let S k,m be the pseudosphere of signature (k, m). We show that L 0 (S k,m ) is a manifold and describe geometrically its canonical contact distribution in terms of the space of oriented geodesics of certain totally geodesic degenerate hypersurfaces in S k,m . Further, we find a contactomorphism with some standard contact manifold, namely, the unit tangent bundle of some pseudo-Riemannian manifold. Also, we express the null billiard operator on L 0 (S k,m ) associated with some simple regions in S k,m in terms of the geodesic flows of spheres.For N the pseudo-Riemannian product of two complete Riemannian manifolds, we give geometrical conditions on the factors for L 0 (N ) to be a manifold and exhibit a contactomorphism with some standard contact manifold.
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