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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, the authors consider the evolution of a compact segment of an analytic curve on the unit tangent bundle of a finite volume hyperbolic $n$-manifold under the geodesic flow.
Abstract: We consider the evolution of a compact segment of an analytic curve on the unit tangent bundle of a finite volume hyperbolic $n$-manifold under the geodesic flow. Suppose that the curve is not contained in a stable leaf of the flow. It is shown that under the geodesic flow, the normalized parameter measure on the curve gets asymptotically equidistributed with respect to the normalized natural Riemannian measure on the unit tangent bundle of a closed totally geodesically immersed submanifold.
Moreover, if this immersed submanifold is a proper subset, then a lift of the curve to the universal covering space $T^1(H^n)$ is mapped into a proper subsphere of the ideal boundary sphere $\partial H^n$ under the visual map. This proper subsphere can be realized as the ideal boundary of an isometrically embedded hyperbolic subspace in $H^n$ covering the closed immersed submanifold.
In particular, if the visual map does not send a lift of the curve into a proper subsphere of $\partial H^n$, then under the geodesic flow the curve gets asymptotically equidistributed on the unit tangent bundle of the manifold with respect to the normalized natural Riemannian measure.
The proof uses dynamical properties of unipotent flows on finite volume homogeneous spaces of SO(n,1).
19 citations
TL;DR: In this paper, a generalization of Lagrangian geometry is proposed, which consists of replacing the tangent bundle by a general tangent manifold, and the Lagrangians by a family of compatible, local, Lagrangeian functions.
Abstract: Lagrange geometry is the geometry of the tensor field defined by
the fiberwise Hessian of a nondegenerate Lagrangian function on
the total space of a tangent bundle. Finsler geometry is the
geometrically most interesting case of Lagrange geometry. In this
paper, we study a generalization which consists of replacing the
tangent bundle by a general tangent manifold, and the Lagrangian
by a family of compatible, local, Lagrangian functions. We give
several examples and find the cohomological obstructions to
globalization. Then, we extend the connections used in Finsler
and Lagrange geometry, while giving an index-free presentation of
these connections.
19 citations
TL;DR: In this article, the Euler characteristic of the accessible region in configuration space is used to determine the topology of a 3-manifold in a magnetic field for the case of 2-degree-of-freedom Hamiltonian systems.
Abstract: Two questions on the topology of compact energy surfaces of natural two degrees of freedom Hamiltonian systems in a magnetic field are discussed. We show that the topology of this 3-manifold (if it is not a unit tangent bundle) is uniquely determined by the Euler characteristic of the accessible region in configuration space. In this class of 3-manifolds for most cases there does not
19 citations
TL;DR: Druţa et al. as discussed by the authors studied the conditions under which a Kahlerian structure (G, J ) of general natural lift type on the tangent bundle TM of a Riemannian manifold (M, g ), studied in [S.I. Cuza] is Einstein.
Abstract: We study the conditions under which a Kahlerian structure ( G , J ) , of general natural lift type on the tangent bundle TM of a Riemannian manifold ( M , g ) , studied in [S. Druţa, V. Oproiu, General natural Kahler structures of constant holomorphic sectional curvature on tangent bundles, An. St. Univ. “Al.I. Cuza” Mat. 53 (2007) 149–166], is Einstein. We found three cases. In the first case the first proportionality factor λ is expressed as a rational function of the first two essential parameters involved in the definition of J and the value of the constant sectional curvature c of the base manifold ( M , g ) . It follows that ( T M , G , J ) has constant holomorphic sectional curvature (Theorem 8). In the second case a certain second degree homogeneous equation in the proportionality factor λ and its first order derivative λ ′ must be fulfilled. After some quite long computations done by using the Mathematica package RICCI for doing tensor computations, we obtain an Einstein Kahler structure only on ( T 0 M , G , J ) ⊂ ( T M , G , J ) , where T 0 M denotes the subset of nonzero tangent vectors to M (Theorem 9). In the last case we obtain that the Kahlerian manifold ( T M , G , J ) cannot be an Einstein manifold.
19 citations
TL;DR: In this article, the differential equations of geodesics of the spacetime tangent bundle are reduced and investigated in the special case of a Riemannian spacetime base manifold.
Abstract: Recent interest in maximal proper acceleration as a possible principle generalizing the theory of relativity can draw on the differential geometry of tangent bundles, pioneered by K. Yano, E. T. Davies, and S. Ishihara. The differential equations of geodesics of the spacetime tangent bundle are reduced and investigated in the special case of a Riemannian spacetime base manifold. Simple relations are described between the natural lift of ordinary spacetime geodesics and geodesics in the spacetime tangent bundle.
18 citations