Showing papers on "Unit tangent bundle published in 1977"

The bundle boundary in some special cases

Abstract: We examine a class of two‐dimensional Lorentz manifolds which are ’’singular’’ in a certain sense. It is shown that, for such a manifold (M, g), the bundle boundary is a single point whose only neighborhood is all of M [the bundle completion of M; see B. G. Schmidt, Gen. Rel. Gravitation 1, 269–80 (1971)]. The four‐dimensional Schwarzschild and Friedmann–Robertson–Walker solutions are then investigated. We show that the bundle completions of these spaces are not Hausdorff.

Structure of symmetric tensors of type (0,2) and tensors of type (1,1) on the tangent bundle

Abstract: The concepts of M-tensor and M-connection on the tangent bundle TM of a smooth manifold M are used in a study of symmetric tensors of type (0, 2) and tensors of type (1, 1) on TM. The constructions make use of certain local frames adapted to an M-connection. They involve extending known results on TM using tensors on M to cases in which these tensors are replaced by M-tensors. Particular attention is devoted to (pseudo-) Riemannian metrics on TM, notably those for which the vertical distribution on TM is null or nonnull, and to the construction of almost product and almost complex structures on TM.

Group contraction in a fiber bundle with Cartan connection

Abstract: A contraction of the structural group with respect to the stability subgroup is performed in a fiber bundle with Cartan connection. The relation of the connections in the original and in the contracted bundle is examined. As an example interesting for physics the contraction of the SO(4,1) de Sitter bundle over space–time to the affine tangent bundle over space–time is discussed with the latter bundle possessing the Poincare group as structural group.

Horocycle flows on certain surfaces without conjugate points

Abstract: We study the topological but not ergodic properties of the horocycle flow {h,} in the unit tangent bundle SM of a complete two dimensional Riemannian manifold M without conjugate points that satisfies the "uniform Visibility" axiom. This axiom is implied by the curvature condition K < c < 0 but is weaker so that regions of positive curvature may occur. Compactness is not assumed. The method is to relate the horocycle flow to the geodesic flow for which there exist useful techniques of study. The nonwandering set QA Q SM for {/j,} is classified into four types depending upon the fundamental group of M. The extremes that sA be a minimal set for {h,} and that Qh admit periodic orbits are related to the existence or nonexistence of compact "totally convex" sets in M. Periodic points are dense in Qh if they exist at all. The only compact minimal sets in Qh are periodic orbits if M is noncompact. The flow (h,) is minimal in S M if and only if M is compact. In general {A,} is topologically transitive in Uh and the vectors in Qh with dense orbits are classified. If the fundamental group of M is finitely generated and Qk = SM then (hi) is topologically mixing in SM. Introduction. Horocycles have played an important role in noneuclidean geometry since its beginning, but horocycle flows on the unit tangent bundle of an orientable surface were evidently studied seriously for the first time by Hedlund and Hopf in the 1930's. The horocycle flow was defined for surfaces of constant negative curvature and was shown to be minimal if M is compact and ergodic if M has finite area. Apparently there was no study of the horocycle flow for the case of an arbitrary orientable, noncompact surface where the nonwandering set of the flow need not be the full unit tangent bundle, SM, of M. In this paper we define and study the horocycle flow on the unit tangent bundle of a more general class of orientable surfaces, the unijorm Visibility surfaces. We consider arbitrary surfaces of this type, both compact and noncompact, and we obtain basic information about the nonwandering set flA Q SM. In particular we classify fiA into four possible types, find criteria for the existence and classification of dense orbits in QA and the existence and

Estimation for jump processes in the tangent bundle of a Riemann manifold

Abstract: A stochastic process is formulated in the tangent bundle of a Riemann manifold where the vector fibre portion of the process is a jump process. Since the tangent spaces change as the process in the base manifold evolves, it is necessary to define a jump process in the fibres of the tangent bundle with respect to the process in the base manifold. An estimation problem is formulated and solved for a process obtained from the jump process in the fibres of the tangent bundle where the observations include the process in the base manifold and the jump times. Since each fibre of the tangent bundle is a linear space, a suitable modification of some results for estimation in linear spaces can be used to solve the aforementioned estimation problem.