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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, extreme values of group-indexed stable random fields for discrete groups acting geometrically on spaces $X$ were studied and the connection between extreme values and the geometric action was mediated by the action of the group $G$ on its limit set equipped with the Patterson-Sullivan measure.
Abstract: We study extreme values of group-indexed stable random fields for discrete groups $G$ acting geometrically on spaces $X$ in the following cases:
1) $G$ acts freely, properly discontinuously by isometries on a CAT(-1) space $X$,
2) $G$ is a lattice in a higher rank Lie group, acting on a symmetric space $X$,
3) $G$ is the mapping class group of a surface acting on its Teichmuller space.
The connection between extreme values and the geometric action is mediated by the action of the group $G$ on its limit set equipped with the Patterson-Sullivan measure. Based on motivation from extreme value theory, we introduce an invariant of the action called extremal cocycle growth which measures the distortion of measures on the boundary in comparison to the movement of points in the space $X$ and show that its non-vanishing is equivalent to finiteness of the Bowen-Margulis measure for the associated unit tangent bundle $U(X/G)$ provided $X/G$ has non-arithmetic length spectrum. As a consequence, we establish a dichotomy for the growth-rate of a partial maxima sequence of stationary symmetric $\alpha$-stable ($0 < \alpha < 2$) random fields indexed by groups acting on such spaces. We also establish analogous results for normal subgroups of free groups.
4 citations
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TL;DR: In this paper, a trivialization of second iterated bundles of a Lie group that preserve lifted group structures is considered, and all possible Poisson, symplectic and Lagrangian reductions of spaces and corresponding dynamics on them are presented.
Abstract: We consider trivializations of second iterated bundles of a Lie group that preserve lifted group structures. With such a trivialization, we elaborate Hamiltonian dynamics on cotangent, Lagrangian dynamics on tangent bundles and, both Hamiltonian and Lagrangian dynamics on Tulczyjew’s symplectic space which is tangent of cotangent bundle of Lie group. We present all possible Poisson, symplectic and Lagrangian reductions of spaces and corresponding dynamics on them. In particular, reduction of Lagrangian dynamics on second iterated tangent bundle includes reduction of dynamics on second order tangent bundle.
4 citations
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TL;DR: In this article, the Lagrange-Poincare equations and equations for the relative equilibria were obtained for a mechanical system with a symmetry describing the motion of two interacting scalar particles on a special Riemannian manifold (the product of the total space of the principal fiber bundle and the vector space).
Abstract: Using the dependent coordinates, the local Lagrange-Poincare equations and equations for the relative equilibria are obtained for a mechanical system with a symmetry describing the motion of two interacting scalar particles on a special Riemannian manifold (the product of the total space of the principal fiber bundle and the vector space) on which a free proper and isometric action of a compact semi-simple Lie group is given As in gauge theories, dependent coordinates are implicitly determined by means of equations representing the local sections of the principal fiber bundle
4 citations
TL;DR: In this article, it was shown that the space of directions at each point in which geodesic rays avoid a sufficiently small neighborhood of a fixed rank 1 vector v∈UM looks very much like a generalized Sierpinski carpet.
Abstract: It is proved that for a Riemannian manifold M with nonpositive sectional curvature and finite volume the space of directions at each point in which geodesic rays avoid a sufficiently small neighborhood of a fixed rank 1 vector v∈UM looks very much like a generalized Sierpinski carpet. We also show for nonpositively curved manifolds M with dim M≥ 3 the existence of proper closed flow invariant subsets of the unit tangent bundle UM whose footpoint projection is the whole of M.
4 citations
TL;DR: In this paper, the authors considered the holomorphic sectional curvature of an almost Hermitian manifold as a differentiable function on the unit tangent bundle of the manifold.
Abstract: Let M = ( M, J, g ) be an almost Hermitian manifold and U ( M )the unit tangent bundle of M . Then the holomorphic sectional curvature H = H ( x ) can be regarded as a differentiable function on U ( M ). If the function H is constant along each fibre, then M is called a space of pointwise constant holomorphic sectional curvature. Especially, if H is constant on the whole U ( M ), then M is called a space of constant holomorphic sectional curvature. An almost Hermitian manifold with an integrable almost complex structure is called a Hermitian manifold. A real 4-dimensional Hermitian manifold is called a Hermitian surface. Hermitian surfaces of pointwise constant holomorphic sectional curvature have been studied by several authors (cf. [2], [3], [5], [6] and so on).
4 citations