Showing papers on "Unit tangent bundle published in 2002"
TL;DR: A detailed and unified presentation of some of the best known results on the geometry of tangent bundles of Riemannian manifolds can be found in this paper, where the main aim of the survey is to present a unified presentation.
158 citations
01 Nov 2002
TL;DR: In this paper, it was shown that there is a hypercomplex structure in the neighbourhood of the zero section of the tangent bundle TX of any complex manifold X with a real-analytic torsion-free connection compatible with the complex structure whose curvature is of type (1, 1).
Abstract: Using twistor techniques we shall show that there is a hypercomplex structure in the neighbourhood of the zero section of the tangent bundle TX of any complex manifold X with a real-analytic torsion-free connection compatible with the complex structure whose curvature is of type (1, 1). The zero section is totally geodesic and the Obata connection restricts to the given connection on the zero section.We also prove an analogous result for vector bundles: any vector bundle with real-analytic connection whose curvature is of type (1, 1) over X can be extended to a hyperholomorphic bundle over a neighbourhood of the zero section of TX.
22 citations
TL;DR: In this paper, the existence of complete Kahler-Einstein metrics with nonpositive Einstein constant and underlying manifold diffeomorphic to the tangent bundle of the (n + 1)-sphere is studied.
Abstract: We give an elementary treatment of the existence of complete Kahler–Einstein metrics with nonpositive Einstein constant and underlying manifold diffeomorphic to the tangent bundle of the (n + 1)-sphere.
19 citations
TL;DR: In this paper, the authors used C∞-singularity theory to describe natural conditions on nonimmersivemappings of a compact orientable 2-manifold into Euclidean 3-space (e.g., the existence of a global limiting tangent bundle).
Abstract: Using C∞-singularity theory, we describe natural conditions on nonimmersivemappings of a compact orientable 2-manifold into Euclidean 3-space (e.g., theexistence of a global limiting tangent bundle). These conditions are weaker thanthose of caustic theory and imply that the image is a CW-complex. In general, the domainmanifold is not homeomorphic to the image CW-complex. We then adapt classical methods to this setting. (This adaptation is simplified by freely moving betweenthe Gauss, Cartan, and Kozul formulations for surface theory.) As an application,we show that part of the extrinsic topology of the image is determined by the firstfundamental form, i.e., a global `theorem egregium'.
15 citations
TL;DR: In this article, the authors developed homological conditions for a manifold to have a unit tangent bundle and applied these criteria to several classical examples: a particle in a potential well, the double spherical pendulum, the Kovalevskaya top, and the N-body problem.
Abstract: When a Hamiltonian system has a Kinetic + Potential structure, the resulting flow is locally a geodesic flow. But there may be singularities of the geodesic structure; so the local structure does not always imply that the flow is globally a geodesic flow. In order for a flow to be a geodesic flow, the underlying manifold must have the structure of a unit tangent bundle. We develop homological conditions for a manifold to have such a structure. We apply these criteria to several classical examples: a particle in a potential well, the double spherical pendulum, the Kovalevskaya top, and the N-body problem. We show that the flow of the reduced planar N-body problem and the reduced spatial 3-body are never geodesic flows except when the angular momentum is zero and the energy is positive.
14 citations
TL;DR: In this article, it was shown that the holonomy group of a Riemannian connection on a k-dimensional Euclidean vector bundle is transitive on the unit sphere bundle whenever the Euler class is spherical.
Abstract: In this note, we show that the holonomy group of a Riemannian connection on a k-dimensional Euclidean vector bundle is transitive on the unit sphere bundle whenever the Euler class is spherical. We extract several consequences from this, among them that this is always the case as long as does not vanish, and the base of the bundle is simply connected and rationally (k + 1)/2-connected.
14 citations
13 citations
Journal Article•
9 citations
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TL;DR: In this paper, the authors compare the spinor bundle of a Riemannian manifold with the bundle obtained by tensoring the Spinor bundles of the Rieman factors in an appropriate way.
Abstract: In this note we compare the spinor bundle of a Riemannian manifold $(M=M_1\times\times M_N,g)$ with the spinor bundles of the Riemannian factors $(M_i,g_i)$ We show, that - without any holonomy conditions - the spinor bundle of $(M,g)$ for a special class of metrics is isomorphic to a bundle obtained by tensoring the spinor bundles of $(M_i,g_i)$ in an appropriate way
6 citations
TL;DR: In this paper, it was shown that the projection of the integral manifold onto the configuration space is a homotopy equivalence, and use this to compute the homology of integral manifolds.
Abstract: The integral manifolds of the N-body problem are the level sets of energy and angular momentum. For positive energy and non-zero angular momentum, all level sets are diffeomorphic to a non-zero level set of angular momentum on the unit tangent bundle of the configuration space. The one complication that arises in attempting to describe this level set explicitly is the degeneracy at the syzygies of the equations that define angular momentum. In this work, we analyze the behavior of the angular momentum near syzygies, and show how to construct local coordinates near the syzygies. In particular, we show that the projection of the integral manifold onto the configuration space \(K\)c is a homotopy equivalence, and use this to compute the homology of the integral manifolds.
6 citations
01 Jan 2002
TL;DR: In this article, the authors deduce general properties of product preserving bundle functors on the category of fibered manifolds and study the prolongation of projectable tangent valued forms with respect to these functors and describe the complete lift of the Froelicher-Nijenhuis bracket.
Abstract: First we deduce some general properties of product preserving
bundle functors on the category of fibered manifolds. Then we
study the prolongation of projectable tangent valued forms with
respect to these functors and describe the complete lift of the
Froelicher-Nijenhuis bracket.
TL;DR: For any > 0, this article constructed an explicit smooth Riemannian metric on the sphere S n, n " 3, that is within! of the round metric and has a geodesic for which the complement of the closure of the corresponding orbit of the geodesIC flow has Liouville measure less than!.
Abstract: For any ! > 0, we construct an explicit smooth Riemannian metric on the sphere S n , n " 3, that is within ! of the round metric and has a geodesic for which the corresponding orbit of the geodesic flow is ! -dense in the unit tangent bundle. Moreover, for any ! > 0, we construct a smooth Riemannian metric on S n ,n " 3, that is within ! of the round metric and has a geodesic for which the complement of the closure of the correspondingorbit of the geodesic flow has Liouville measure less than ! .
TL;DR: In this article, the geometric behavior of the normal bundle of a submanifold M of a Riemannian manifold was studied and the relation between the minimality of the bundle and the Maslov forms with respect to a suitable connection of the pair was analyzed.
Abstract: We study the geometric behavior of the normal bundle T
⊥
M of a submanifold M of a Riemannian manifold . We compute explicitely the second fundamental form of T
⊥
M and look at the relation between the minimality of T
⊥
M and M. Finally we show that the Maslov forms with respect to a suitable connection of the pair (T
⊥
M, are null.
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.
01 Jan 2002
TL;DR: In this article, the sectional curvature of tangent sphere bundles over locally symmetric Riemannian manifolds has been studied, and it has been shown that the converse of Theorem 1 also holds.
Abstract: The authors proved a theorem about the sectional curvature of tangent sphere bundles over locally symmetric Riemannian manifolds (see Theorem A below). After a slight generalization of this theorem (Theo- rem 1) we prove several results which give strong support of the conjecture that the converse of Theorem 1 also holds. The problem still remains open, in general.
TL;DR: In this article, the authors consider a product preserving functor F of order r and a connection Γ on a manifold M and introduce horizontal lifts of tensor fields and linear connections from M to F(M) with respect to Γ.
Abstract: In this paper we consider a product preserving functor F of order r and a connection Γ of order r on a manifold M. We introduce horizontal lifts of tensor fields and linear connections from M to F(M) with respect to Γ. Our definitions and results generalize the particular cases of the tangent bundle and the tangent bundle of higher order.
TL;DR: In this article, the existence of adapted complex structures for real-analytic Riemannian manifolds is examined under the point view of complexifications of geodesic flows, and a sufficient criterion is given as to when a domain in the tangent bundle of the manifold is a maximal domain of definition of an adapted complex structure.
TL;DR: In this paper, it was shown that a class of regular centers, so that quasi-smoothness is preserved by blowing-up, can be enlarged and characterized by means of their jacobian ideal and some geometrical conditions.
Abstract: We study a notion of quasi-smoothness which makes possible to associate a canonical tangent bundle to certain arithmetical schemes (see [V1]). In this context we prove that we can associate a jacobian ideal to any irreducible subscheme. A class of regular centers, so that quasi-smoothness is preserved by blowing-up, will be enlarged and characterized by means of their jacobian ideal and some geometrical conditions, improving the results appearing in [V1].
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TL;DR: In this article, the authors characterize Kaehler manifolds with trivial logarithmic tangent bundle (with respect to a divisor D) as a class of certain compatifications of complex semi-tori.
Abstract: We characterize Kaehler manifolds with trivial logarithmic tangent bundle (with respect to a divisor D) as a class of certain compatifications of complex semi-tori.
01 Jan 2002
TL;DR: In this article, the conformal almost symplectic structure in the tangent bundle is defined and the set of all general conformal-almost-symmetric d-linear connections on TM is determined.
Abstract: The present paper deals with the conformal almost symplectic structure on TM. Starting from the notion of conformal almost symplectic structure in the tangent bundle, we define the notion of general conformal almost symplectic d-linear connection and respective conformal almost symplectic d-linear connection with respect to a conformal almost symplectic structure Â, corresponding to the 1-forms ω and ω in TM . We determine the set of all general conformal almost symplectic d-linear connections on TM , in the case when the nonlinear connection is arbitrary and we find important particular cases.
TL;DR: In this paper, the authors studied the behavior of the quasi-smoothness condition in the case of arithmetical regular surfaces and showed that it is possible to associate a canonical tangent bundle to certain arithmically regular schemes.
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27 Oct 2002
TL;DR: In this article, it was shown that on a 4-manifold M endowed with a spin-structure induced by an almost-complex structure, a self-dual spinor field φ ∈ Γ(W) is the same as a bundle morphism φ : TM → TM acting on the fiber by selfdual conformal transformations, such that the Clifford multiplication is just the evaluation of φ on tangent vectors.
Abstract: We show that, on a 4-manifold M endowed with a spin -structure induced by an almost-complex structure, a self-dual (= positive) spinor field φ ∈ Γ(W) is the same as a bundle morphism φ : TM → TM acting on the fiber by self-dual conformal transformations, such that the Clifford multiplication is just the evaluation of φ on tangent vectors, and that the squaring map σ : W → Λ acts by pulling-back the fundamental form of the almost-complex structure. We use this to detect Kähler and symplectic
01 Jan 2002
TL;DR: In this article, the authors consider a product preserving functor F of order r and a connection of order Ron a manifold M. They introduce horizontal lifts of tensor elds and linearconnections from MtoF(M)withrespectto.
Abstract: In this paper we consider a product preserving functor F of order r and a connection of order ron a manifold M. We introduce horizontal lifts of tensor elds and linearconnectionsfrom MtoF(M)withrespectto. Ourdenitionsandresultsgeneralize the particular cases of the tangent bundle and the tangent bundle of higher order.
TL;DR: For a compact riemannian manifold of negative curvature, the geodesic foliation of its unit tangent bundle is independent of the negatively curved metric, up to Holder bicontinuous homeomorphism as mentioned in this paper.
Abstract: For a compact riemannian manifold of negative curvature, the geodesic foliation of its unit tangent bundle is independent of the negatively curved metric, up to Holder bicontinuous homeomorphism. However, the riemannian metric defines a natural transverse measure to this foliation, the Liouville transverse measure, which does depend on the metric. For a surface S, we show that the map which to a hyperbolic metric on S associates its Liouville transverse measure is differentiable, in an appropriate sense. Its tangent map is valued in the space of transverse Holder distributions for the geodesic foliation.