Showing papers on "Unit tangent bundle published in 1993"

The orthogonal spectrum of a hyperbolic manifold

Abstract: Introduction. In this paper, we investigate the geometry of embedded hypersurfaces in hyperbolic manifolds of dimension greater than one. Our focus is on hypersurfaces that are either totally geodesic, horospherical (quotients of horospheres), or the boundary of a hyperbolic ball. These hypersurfaces have a natural decomposition into two pieces; one of which corresponds to base points of normal rays that satisfy a specified crossing condition, and the other piece being a disjoint union of embedded discs. We show that these embedded discs are in one to one correspondence with a certain set of lengths of orthogonals (the orthogonal spectrum) emanating from the hypersurface. In fact, the area of each disc is a monotonic function of the corresponding length in the spectrum. Thus the area (induced volume) of the hypersurface is intimitely related to the orthogonal spectrum (see theorem 1.1 for a precise statement). In special cases, (corollary 1.2) the piece of the hypersurface corresponding to the base points of normal rays is related to the limit set of the discrete isometry group that represents the manifold. As a consequence, we show that the measure of the limit set of this group is zero if and only if the area of the piece consisting of base points of normal rays is zero. The next theorem (theorem 1.3) relates the orthogonal spectrum of a closed embedded totally geodesic hypersurface with the lengths of geodesic loops based at some point of the ambient manifold. In particular, by applying a result of Sullivan, we can show that the geodesic flow acts ergodically on the unit tangent bundle of the manifold if and only if the sum of the areas of the discs embedded in the hypersurface diverges (corollary 1.4).

Complex Structures on the Tangent Bundle of Riemannian Manifolds

Abstract: It is well known that any (paracompact) differentiable manifold M has a complexification, i.e., a complex manifold X ⊃ M, dimc X = dimℝ M, such that M is totally real in X (see Ref. 8). It is also known that a small neighborhood U of M in X is diffeomorphic to the tangent bundle TM of M. Thus, the tangent bundle TM of any differentiable manifold carries a complex manifold structure. This complex structure is, of course, not unique. One way of finding a “canonical” complex structure is to endow M with some extra structure and require that the complex structure on TM interact with the structure of M. Here we consider smooth (meaning infinitely differentiable) Riemannian manifolds M. When M = ℝ, there is a natural identification Tℝ ≅ ℂ given by $${{T}_{\sigma }}\mathbb{R} \mathrel\backepsilon \tau \frac{\partial }{{\partial \sigma }} \leftrightarrow \sigma + i\tau \in \mathbb{C},$$ (1.1) and this endows Tℝ with a complex structure. In (1.1) σ denotes the coordinate on R. This coordinate depends on the algebraic structure of the identification (1.1); however, the complex structure on Tℝ depends only on the metric of ℝ. In other words, an isometry of ℝ induces a biholomorphic mapping on Tℝ.

Complex spacetime tangent bundle

Abstract: It is demonstrated that the spacetime tangent bundle, in the case of a Finsler spacetime, is complex, provided that the gauge curvature field vanishes. This is accomplished by determining the conditions for the vanishing of the Nijenhuis tensor in the anholonomic frame adapted to the spacetime connection.

Spacetime tangent bundle with torsion

Abstract: It is demonstrated explicitly that the bundle connection of the Finslerspacetime tangent bundle can be made compatible with Cartan's theory of Finsler space by the inclusion of bundle torsion, and without the restriction that the gauge curvature field be vanishing. A component of the contorsion is made to cancel the contribution of the gauge curvature field to the relevant component of the bundle connection. Also, it is shown that the bundle manifold remains almost complex, and that the almost complex structure can be made to have a vanishing covariant derivative if additional conditions on the torsion are satisfied. However, the Finsler-spacetime tangent bundle remains complex only if the gauge curvature field vanishes.

Prolongations of f-structure to.the tangent bundle of order 2

Abstract: A study of prolongations of F-structure to the tangent bundle of order 2 has been presented.

A characterization of complete Riemannian manifolds minimally immersed in the unit sphere

Abstract: Let M n be an n -dimensional Riemannian manifold minimally immersed in the unit sphere S n+p (1) of dimension n + p . When M n is compact, Chern, do Carmo and Kobayashi [1] proved that if the square ‖ h ‖ 2 of length of the second fundamental form h in M n is not more than , then either M n is totally geodesic, or M n is the Veronese surface in S 4 (1) or M n is the Clifford torus . In this paper, we generalize the results due to Chern, do Carmo and Kobayashi [1] to complete Riemannian manifolds.

L'espace des plongements d'un arc dans une surface

Abstract: We prove that the space of embeddings of an arc into a surface without boundary M is homeomorphic to the product U(M) × l 2 , where U(M) is the unit tangent bundle of M

Dynamical systems, Santiago de Chile, 1990

Abstract: Cantor sets, numerical invariants and Perron Frobenius theory A modulus of stability for the Sil'nikov theorem Heteroclinic orbits for singularly perturbed differential-difference equations Simultaneous Hopf bifurcations at the origin and infinity for cubic systems Nondensity of ()-stable endomorphisms and rough ()-stabilities for endomorphisms Supplement to homoclinic doubling bifurcation in vector fields On stable and unstable sets The fundamental theorem of Sinai-Chernov for dynamical systems with singularities Singular perturbations of autonomous ordinary differential equations and heteroclinic bifurcation Expansivity and length expansivity for geodesic flows on surfaces Rigidity of the C1-centralizer of bidimensional diffeomorphisms Poisson brackets and dynamics A note on algebraic solutions of foliations in dimension two Lines of curvature near hyperbolic principal cycles Arithmetic, geometry and dynamics in the unit tangent bundle of the modular orbifold Three dimensional expansive homeomorphisms Introduction to the ergodic theory of plan billiards

On Dynamical Systems and the Minimal Surface Equation

Abstract: The relationship between the theory of dynamical systems and differential geometry is a long-standing and profound one. It has largely been focused around the geodesic flow on the unit tangent bundle of a Riemannian manifold. One may recall early work by Poincare and Birkhoff for the case of convex surfaces, Morse theory, contributions by Hadamard, and, more recently, by Anosov, in the case of negatively curved manifolds—to register the decisive influence that has been asserted by this particular example in the development of the general theory of dynamical systems. Conversely, analysis of the geodesic flow has been useful to study problems in Riemannian geometry; one may point to the recent successful structure theory for Hadamard manifolds by Ballmann, Eberlein, Gromov, Schroeder, and others. Here we will be concerned with a different, more unexpected role that dynamical systems have recently played in the very active field of partial differential equations that arise in Riemannian geometry, notably the minimal surface equation. Briefly, in the presence of a suitably chosen Lie group G of isometries, one studies the G-invariant solutions. A simple case of this idea goes back to Delaunay’s classification of rotationally invariant constant mean curvature surfaces [D]; more recently, the celebrated work of Bombieri, de Giorgi, and Giusti on the Bernstein problem may also be viewed in this context [BGG]. The ideas were developed in a more systematic way as a general program of “equivariant geometry” in a seminal paper by W.Y. Hsiang and B. Lawson [HL], and has since been developed, especially by Hsiang, to solve some long-standing open problems in Riemannian geometry [HI, H2, H3].