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Showing papers on "Unit tangent bundle published in 1994"


Book ChapterDOI
TL;DR: Tangent spaces of a sub-Riemannian manifold are themselves sub-riemannians as mentioned in this paper, and they come with an algebraic structure: nilpotent Lie groups with dilations.
Abstract: Tangent spaces of a sub-Riemannian manifold are themselves sub-Riemannian manifolds. They can be defined as metric spaces, using Gromov’s definition of tangent spaces to a metric space, and they turn out to be sub-Riemannian manifolds. Moreover, they come with an algebraic structure: nilpotent Lie groups with dilations. In the classical, Riemannian, case, they are indeed vector spaces, that is, abelian groups with dilations. Actually, the above is true only for regular points. At singular points, instead of nilpotent Lie groups one gets quotient spaces G/H of such groups G.

749 citations


Journal ArticleDOI
Michael Struwe1
TL;DR: In this article, the Yang-Mills heat flow in a vector bundle over a compact Riemannian four-manifold for given initial connection of finite energy is established.
Abstract: Global existence and uniqueness is established for the Yang-Mills heat flow in a vector bundle over a compact Riemannian four-manifold for given initial connection of finite energy. Our results are analogous to those valid for the evolution of harmonic maps of Riemannian surfaces.

78 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that the volume of ImM, calculated with respect to a Riemannian metric induced naturally by the Finsler structure, is in general a function of x.
Abstract: Motivated by some issues which enter into the Gauss-Bonnet-Chern theorem in Finsler geometry, this paper studies the unit tangent sphere (or indicatrix) Ix M at each point x of a Pinsler manifold M. We demonstrate that the volume of ImM, calculated with respect to a Riemannian metric induced naturally by the Finsler structure, is in general a function of x. This contrasts sharply with the situation in Riemannian geometry. We also express the derivative of such volume function in terms of the second curvature tensor of the Chern connection. In particular, we find that this function is constant on Landsberg spaces (though that constant need not be equal to the value taken by Riemannian manifolds).

43 citations


Journal ArticleDOI
Gerhard Knieper1
TL;DR: In this article, it was shown that the spherical mean of functions on the unit tangent bundle of a compact manifold of negative curvature converges to a measure containing a vast amount of information about the asymptotic geometry of those manifolds.
Abstract: We show that the spherical mean of functions on the unit tangent bundle of a compact manifold of negative curvature converges to a measure containing a vast amount of information about the asymptotic geometry of those manifolds. This measure is related to the unique invariant measure for the strong unstable foliation, as well as the Patterson-Sullivan measure at infinity. It turns out to be invariant under the geodesic flow if and only if the mean curvature of the horospheres is constant. We use this measure in the study of rigidity problems.

40 citations


Journal ArticleDOI
TL;DR: In this paper, the tangent Lie algebroid is shown to be equivalent to a certain Poisson structure on a smooth vector bundle A to (pi over)Q, which gives rise to a Lie algebraic structure on the bundle TA to (T pi over)TQ.
Abstract: This paper shows that a Lie algebroid structure on a smooth vector bundle A to ( pi over)Q gives rise to a Lie algebroid structure on the bundle TA to (T pi over)TQ, called the tangent Lie algebroid. The analysis uses global arguments. A Lie algebroid A is equivalent to a certain Poisson structure on A*, and the tangent bundle of any Poisson manifold has a tangent Poisson structure. The tangent Poisson structure on TA* is then dualized to produce the tangent Lie algebroid structure on TA. Local calculations are used, and formulae for local brackets are given.

39 citations



Journal ArticleDOI
01 Nov 1994
TL;DR: In this paper, it was shown that the geodesic flow on the unit tangent bundle SM of a Riemannian manifold is ergodic in the sense that its Hausdorff dimension is equal to or greater than that of the manifold itself.
Abstract: Let M be a Riemannian manifold of constant negative curvature and finite Riemannian volume. It is well-known that the geodesic flow on the unit tangent bundle SM of M is ergodic. In particular, it follows that for almost all ( p , v )∈ SM , where p ∈ M and v is a unit tangent vector at p , the geodesic through p in the direction of v is dense in M . A theorem of Dani [Dl] says that the set of all ( p, v )∈ SM for which the corresponding geodesic is bounded (namely those with compact closure in M ) is ‘large’ in the sense that its Hausdorff dimension is equal to that of the unit tangent bundle itself. In fact, Dani generalized this result to a more general algebraic situation (cf. [D2]).

19 citations




BookDOI
01 Jan 1994
TL;DR: In this paper, Ghione et al. investigated the restriction of the tangent bundle of IP to a curve X ⊂ IP and showed that the restricted bundle is semistable and moreover simple if g ≥ 2 if the degree is high with respect to the genus.
Abstract: The purpose of this paper is to investigate the restriction of the tangent bundle of IP to a curve X ⊂ IP. The corresponding question for rational curves was investigated by L. Ramella [7] and F. Ghione, A. Iarrobino and G. Sacchiero [2] in the case of rational curves. Let us also mention that D. Laksov [6] proved that the restricted tangent bundle of a projectivly normal curve does not split unless the curve is rational. We will show the following theorem (See 3.1): Theorem In the variety of smooth connected space curves of genus g ≥ 1 and degree d ≥ g+ 3 there exists a nonempty dense open subset where the restricted tangent bundle is semistable and moreover simple if g ≥ 2 If the degree is high with respect to the genus (d > 3g), we get a postulation formula for the strata with a given Harder-Narasimhan polygon, following results of R. Hernandez [5]. In case of plane curves the situation is simpler due to Theorem If X is a smooth plane curve of degree d, the restricted tangent bundle is stable for d ≥ 3, of splitting type (3, 3) for a conic, and of splitting type (2, 1) for a line. Proof: (following D. Huybrechts) We denote by E the tangent bundle of IP 2 twisted by OIP 2(−1). We first suppose that d > 2. We use the facts: 1. E is stable, c1(E) = 1 and c2(E) = 1.

10 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that three different notions of regularity for the stable foliation on the unit tangent bundle of a compact manifold of negative curvature are equivalent, and that if is a time-preserving conjugacy of geodesic flows of such manifolds M, N then the Lyapunov exponents at corresponding periodic points of the flows coincide.
Abstract: It is shown that three different notions of regularity for the stable foliation on the unit tangent bundle of a compact manifold of negative curvature are equivalent. Moreover if is a time-preserving conjugacy of geodesic flows of such manifolds M , N then the Lyapunov exponents at corresponding periodic points of the flows coincide. In particular Δ also preserves the Lebesgue measure class.

Journal ArticleDOI
Abstract: In the Finsler-spacetime tangent bundle, a simple solution is determined to the torsion relations that were obtained previously to maintain (1) compatibility with Cartan's theory of Finsler space, (2) the almost complex structure, and (3) the vanishing of the covariant derivative of the almost complex structure.

Posted Content
TL;DR: In this article, the horizontal splitting of the vertical tangent bundle of a composite fibred manifold is studied, and the splitting defines the modified covariant differential and implies the special fashion of Lagrangian densities of field models on composite manifolds.
Abstract: In classical field theory, the composite fibred manifolds Y -> Z -> X provides the adequate mathematical formulation of gauge models with broken symmetries, e.g., the gauge gravitation theory. This work is devoted to connections on composite fibred manifolds. In particular, we get the horizontal splitting of the vertical tangent bundle of a composite fibred manifold, besides the familiar one of its tangent bundle. This splitting defines the modified covariant differential and implies the special fashion of Lagrangian densities of field models on composite manifolds. The spinor composite bundles are examined.

Dissertation
01 Jan 1994
TL;DR: In this article, a set of geodesics on orientable Riemannian surfaces with geodesic boundary and negative Euler characteristic was shown to be equivalent to a subset of the unit tangent bundle of the surface with the arbitrary Riemanian metric and the standard hyperbolic metric on this surface.
Abstract: This thesis consists of two independent chapters Both present results in the field of dynamical systems In the first chapter we study abstract adding machines and their occurrence in unimodal maps of the interval For unimodal maps with no aperiodic homtervals we characterize completely when adding machines occur We also discuss their importance in relation to the boundary of positive topological entropy for two-parameter families of diffeomorphisms of the disc In the second chapter we prove existence of a distinguished set of geodesics on orientable Riemannian surfaces with geodesic boundary and negative Euler characteristic This result allows us to construct a semi-equivalence between a subset of the unit tangent bundle of the surface with the arbitrary Riemannian metric and the unit tangent bundle given by the standard hyperbolic metric on this surface The result is analogous to one of Morse [Ml] for surfaces without boundary We give a new proof of Morse’s result using a method similar to the proof of our new result

Book ChapterDOI
01 Jan 1994
TL;DR: The geometry of the total space of the tangent bundle to a smooth manifold is very rich and contains a lot of geometrical objects of theoretical interest and of a great importance in constructing of various geometry models useful in Physics.
Abstract: The geometry of the total space of the tangent bundle to a smooth manifold is very rich. It contains a lot of geometrical objects of theoretical interest and of a great importance in constructing of various geometrical models useful in Physics. For the similar reasons the geometry of the total space of a vector bundle was intensively studied in the last fifteen years.

Journal Article
TL;DR: In this paper, the authors apply a result of Nagano to prove that an integrable almost tangent manifold M endowed with a vector field satisfying similar properties to those satisfied by the canonical vector field of a vector bundle admits a unique vector bundle structure such that M is isomorphic to a tangent bundle.
Abstract: We apply a result of Nagano to prove that an integrable almost tangent manifold M endowed with a vector field satisfying similar properties to those satisfied by the canonical vector field of a vector bundle admits a unique vector bundle structure such that M is isomorphic to a tangent bundle. Thus we obtain a characterization of tangent bundles. This characterization was obtained by Crampin et al. and Filippo et al. in a different way. We also extend the result to stable tangent bundles. An application to reduction of degenerate autonomous and non-autonomous Lagrangian systems is given.

01 Jan 1994
TL;DR: In this paper, it was shown that the s tandard contact metric structure on the tangent sphere bundle is locally symmetric if and only if the base manifold is a surface of constant Gaussian curvature 0 or -t--1.
Abstract: In [2] one of the authors showed that the s tandard contact metric structure on the tangent sphere bundle is locally symmetric if and only if the base manifold is fiat or of dimension 2 and of constant curvature +1. In this paper we show that this structure is conformally flat if and only if the base manifold is a surface of constant Gaussian curvature 0 or -t--1. In the final section of the paper we give an additional result on conformally flat contact metric manifolds.

Journal ArticleDOI
TL;DR: Alamo and Gbmez as discussed by the authors proved that a G-invariant neighbourhood of the singular set C in a manifold M is completely determined by the G-vector bundle restriction of the tangent bundle of M to Z.

Journal ArticleDOI
TL;DR: In this paper, the authors studied the compact Kahler manifolds whose tangent bundles are numerically effective and whose anti-Kodaira dimensions are equal to one, and they proved that κ(−K ) = 1 if and only if there exists a finite etale coverY→X such thatY≅ℙ1×A, where A is a complex torus.
Abstract: In this paper, we study the compact Kahler manifolds whose tangent bundles are numerically effective and whose anti-Kodaira dimensions are equal to one LetX be a compact Kahler manifold with nef tangent bundle and semiample anti-canonical bundle We prove that κ(−K X )=1 if and only if there exists a finite etale coverY→X such thatY≅ℙ1×A, whereA is a complex torus As a consequence, we are able to improve upon a result of T Fujiwara [3, 4]