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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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01 Jan 2006
TL;DR: In this paper, a new equation with respect to a unit vector field on Riemannian manifold M such that its solution defines a totally geodesic submanifold in the unit tangent bundle with Sasakian metric was presented.
Abstract: We present a new equation with respect to a unit vector field on Riemannian manifold M such that its solution defines a totally geodesic submanifold in the unit tangent bundle with Sasakian metric and apply it to some classes of unit vector fields. We introduce a class of covariantly normal unit vector fields and prove that within this class the Hopf vector field is a unique global one with totally geodesic property. For the wider class of geodesic unit vector fields on a sphere we give a new necessary and sufficient condition to generate a totally geodesic submanifold in T1S.
4 citations
TL;DR: In this article, it was shown that the submanifolds determined by the left and right invariant sections minimize volume in their homology classes, and that the resulting vector bundle over S3 with the Sasaki metric has as well no parallel unit sections.
Abstract: Gluck and Ziller proved that Hopf vector fields on S3 have minimum volume among all unit vector fields. Thinking of S3 as a Lie group, Hopf vector fields are exactly those with unit length which are left or right invariant, and TS3 is a trivial vector bundle with a connection induced by the adjoint representation. We prove the analogue of the stated result of Gluck and Ziller for the representation given by quaternionic multiplication. The resulting vector bundle over S3, with the Sasaki metric, has as well no parallel unit sections. We provide an application of a double point calibration, proving that the submanifolds determined by the left and right invariant sections minimize volume in their homology classes.
4 citations
TL;DR: In this article, the authors studied the harmonicity properties of the canonical projection π and π 1 to Riemannian g-natural metrics of Kaluza-Klein type.
Abstract: Let (TM, G) and $${(T_1 M,\tilde G)}$$
respectively denote the tangent bundle and the unit tangent sphere bundle of a Riemannian manifold (M, g), equipped with arbitrary Riemannian g-natural metrics. After studying the geometry of the canonical projections π : (TM, G) → (M, g) and $${\pi_1:(T_1 M,\tilde G) \rightarrow (M,g)}$$
, we give necessary and sufficient conditions for π and π
1 to be harmonic morphisms. Some relevant classes of Riemannian g-natural metrics will be characterized in terms of harmonicity properties of the canonical projections. Moreover, we study the harmonicity of the canonical projection $${\Phi:(TM-\{0\},G)\to (T_1 M,\tilde G)}$$
with respect to Riemannian g-natural metrics $${G,\tilde G}$$
of Kaluza–Klein type.
4 citations
TL;DR: In this article, it was shown that if r l and r 2 are foliations which are C k close (k >~ 2), and if X1 and X2 are vector fields preserving rl and r2 respectively, with the same singular set, then the residues determined by rl, X, are close to those determined by R2, X2.
Abstract: In this section we show that if r l and r2 are foliations which are C k close (k >~ 2), and if X1 and X2 are vector fields preserving rl and r2 respectively, with the same singular set, and if X~ is C k close to X2, then the residues determined by rl, X , are close to those determined by r2,X2. Let M be a smooth, n dimensional, oriented manifold with tangent bundle TM. For any bundle E over M we denote the space of smooth sections of E by C'(E). Each element of Co(APTM), the subspace of C\"*(APTM) consisting of non-zero sections, determines a smooth p dimensional subbundle of TM, and every such subbundle is so obtained. Denote by Fq(M) the subset of Co(A n-q TM) consisting of those sections which determine oriented foliations. The space Co(A n-q TM) has a natural C k topology on it (the topology of uniform C k convergence on compact sets) and so it induces a topology on F q (M), the C k topology. Let r E Fq(M) and X a F vector field for r . We assume that Sing X = {x E MIX(x) E rx is a connected leaf N of r , and that M is an R q bundle over N.
4 citations
TL;DR: In this paper, it was shown that there exists a family of Riemannian metrics on the tangent bundle of a two-sphere, which induces metrics of constant curvature on its unit tangent bundles.
Abstract: We show that there exists a family of Riemannian metrics on the tangent bundle of a two-sphere, which induces metrics of constant curvature on its unit tangent bundle. In other words, given such a metric on the tangent bundle of a two-sphere, the Hopf map is identified with a Riemannian submersion from the universal covering space of the unit tangent bundle, equipped with the induced metric, onto the two-sphere. A hyperbolic counterpart dealing with the tangent bundle of a hyperbolic plane is also presented.
4 citations