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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 article, an effective equidistribution of closed horospheres in the unit tangent bundle of a geometrically finite hyperbolic 3-manifold of infinite volume was shown.
Abstract: The main result of this paper is an effective count for Apollonian circle packings that are either bounded or contain two parallel lines. We obtain this by proving an effective equidistribution of closed horospheres in the unit tangent bundle of a geometrically finite hyperbolic 3-manifold of infinite volume, whose fundamental group has critical exponent bigger than 1. We also discuss applications to Affine sieves. Analogous results for surfaces are treated as well.
5 citations
TL;DR: In this paper, it was shown that the tangent bundle of Z is stable and a vanishing result for the top cohomology of the twisted holomorphic differential forms on Z was proved.
Abstract: Let M be a complex projective Fano manifold whose Picard group is isomorphic to Z and the tangent bundle TM is semistable. Let Z ⊂ M be a smooth hypersurface of degree strictly greater than degree(TM)(dimC Z−1)/(2 dimC Z−1) and satisfying the condition that the inclusion of Z in M gives an isomorphism of Picard groups. We prove that the tangent bundle of Z is stable. A similar result is proved also for smooth complete intersections in M . The main ingredient in the proof is a vanishing result for the top cohomology of the twisted holomorphic differential forms on Z.
5 citations
TL;DR: In this paper, the authors used dynamical approach to study parabolic fixed points of Kleinian groups Γ ⊂ Iso(ℍn) with constant negative curvature.
Abstract: We use dynamical approach to study parabolic fixed points of Kleinian groups Γ ⊂ Iso(ℍn). Let ℋ be the horospherical foliation on the unit tangent bundle SM of manifold M = Γ\ℍn with constant negative curvature. We construct examples Γ ⊂ Iso(ℍ4) which show that horosphere based at parabolic fixed point w ∈ ∂ℍ4 can project to leaf ℋx ⊂ SM of complicated structure: it can be locally closed and not closed; not locally closed and non-dense in the non-wandering set Ω+ ⊂ SM of ℋ; dense in Ω+ (this is equivalent to w being a horospherical limit point). Using the natural duality, one gets the corresponding examples of Γ-orbits on the light cone. We give an elementary proof of the fact that conical limit point w ∈ ∂ℍn cannot be a parabolic fixed point.
5 citations
TL;DR: The tangent bundle of a differentiable manifold is an important invariant of a manifold as mentioned in this paper, which is determined neither by the topological structure nor by the homotopy type of the manifold.
Abstract: The tangent bundle of a differentiable manifold is an important invariant of a differentiable structure. It is determined neither by the topological structure nor by the homotopy type of a manifold. But in some cases tangent bundles depend only on the homotopy types of manifolds.
5 citations
01 Jan 2000
TL;DR: In this paper, the Bott connection is defined as a partial connection defined by a splitting on the fundamental sequence of vector bundles (see the definition below), which is defined by the splitting of the vector bundles in complex Finsler geometry.
Abstract: In the present paper, we shall investigate connections theory in complex Finsler geometry. The basic tool in this paper is the so-called Bott connection which is a partial connection defined by a splitting on the fundamental sequence of vector bundles (see the definition below). Let : E ! M be a holomorphic vector bundle over a complex manifold. We denote by TE and TM the holomorphic tangent bundles of E and M respectively. Moreover we denote by TE/M the relative tangent bundle of the holomorphic projection . Then we get the fundamental sequence of vector bundles: O ! TE/M i ! TE d ! TM ! O.
5 citations