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Unit tangent bundle
About: Unit tangent bundle is a research topic. Over the lifetime, 1056 publications have been published within this topic receiving 15845 citations.
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TL;DR: In this paper, it was shown that a measure on the unit tangent bundle of a constant positive curvature space is not uniquely determined by its projection to the positive curvatures.
Abstract: Let $S^{n}$ be the $n$-sphere of constant positive curvature. For $n \geq 2$, we will show that a measure on the unit tangent bundle of $S^{2n}$, which is even and invariant under the geodesic flow, is not uniquely determined by its projection to $S^{2n}$.
1 citations
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TL;DR: In this article, it was shown that the tangent bundle of a complex linear algebraic group is stable with respect to every polarization on the group and that the group can be compactified to a simple adjoint type.
Abstract: Let G be a complex linear algebraic group which is simple of adjoint type. Let G be the wonderful compactification of G. We prove that the tangent bundle of G is stable with respect to every polarization on G.
1 citations
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TL;DR: A complete list of left invariant unit vector fields on three-dimensional Lie groups with the left-invariant metric that generate a totally geodesic submanifold in the unit tangent bundle of a group with the Sasaki metric is given in this paper.
Abstract: We give a complete list of those left invariant unit vector fields on three-dimensional Lie groups with the left-invariant metric that generate a totally geodesic submanifold in the unit tangent bundle of a group with the Sasaki metric. As a result, each class of three-dimensional Lie groups admits the totally geodesic unit vector field. From geometrical viewpoint, the field is either parallel or characteristic vector field of a natural almost contact structure on the group.
1 citations
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TL;DR: In this article, it was shown that for the set of geodesics corresponding to the ideal class group of a real quadratic field, the volume has linear asymptotics in terms of the total length of the geodesic flow.
Abstract: Periodic geodesics on the modular surface correspond to periodic orbits of the geodesic flow in its unit tangent bundle $\mathrm{PSL}_2(\mathbb{Z})\backslash\mathrm{PSL}_2(\mathbb{R})$. The complement of any finite number of orbits is a hyperbolic $3$-manifold, which thus has a well-defined volume. We present strong numerical evidence that, in the case of the set of geodesics corresponding to the ideal class group of a real quadratic field, the volume has linear asymptotics in terms of the total length of the geodesics. This is not the case for general sets of geodesics
1 citations
TL;DR: For manifolds with geodesic constants of motion, the authors showed that almost all Laplacian eigenfunctions become equidistributed on these submanifolds.
Abstract: For manifolds with geodesic flow that is ergodic on the unit tangent bundle, the Quantum Ergodicity Theorem implies that almost all Laplacian eigenfunctions become equidistributed as the eigenvalue goes to infinity. For a locally symmetric space with a universal cover that is a product of several upper half-planes, the geodesic flow has constants of motion so it cannot be ergodic. It is, however, ergodic when restricted to the submanifolds defined by these constants. Accordingly, we show that almost all eigenfunctions become equidistributed on these submanifolds.
1 citations