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.

Tangent Bundle of the Hypersurfaces in a Euclidean Space

Abstract: We consider an immersed orientable hypersurface f : M ! R n+1 of the Euclidean space (f an immersion), and observe that the tan- gent bundle TM of the hypersurface M is an immersed submanifold of the Euclidean space R 2n+2 . Then we show that in general the induced metric on TM is not a natural metric and obtain expressions for the horizontal and vertical lifts of the vector fields on M. We also study the special case in which the induced metric on TM becomes a natural metric and show that in this case the tangent bundle TM is trivial.

Twistor bundle theory and its application

Abstract: Over an oriented even dimensional Riemannian manifold(M 2m ,ds2 ), in terms of the Levi-Civita connection form Ω and the canonical form Θ on the bundle of positive orthonormal frames, we give a detailed description of the twistor bundle Гm = SO(2m)/U(m)↪ J +(@#@ M,ds2 ) →M. The integrability on an almost complex structureJ compatible with the metric and the orientation, is shown to be equivalent to the fact that the corresponding cross section of the twistor bundle is holomorphic with respect toJ and the canonical almost complex structureJ 1 onJ +(M,ds2 ), by using moving frame theory. Moreover, for various metrics and a fixed orientation onM, a canonical bundle isomorphism is established. As a consequence, we generalize a celebrated theorem of LeBrun.

Notes on the tangent bundle with deformed complete lift metric

Abstract: In this paper, our aim is to study some properties of the tangent bundle with a deformed complete lift metric.

The Lagrange-Poincaré equations for a mechanical system with symmetry on the principal fiber bundle over the base represented by the bundle space of the associated bundle

Abstract: The Lagrange--Poincare equations for a mechanical system which describes the interaction of two scalar particles that move on a special Riemannian manifold, consisting of the product of two manifolds, the total space of a principal fiber bundle and the vector space, are obtained. The derivation of equations is performed by using the variational principle developed by Poincare for the mechanical systems with a symmetry. The obtained equations are written in terms of the dependent variables which, as in gauge theories, are implicitly determined by means of equations representing the local sections of the principal fiber bundle.

Hopf conjecture holds for analytic, k -basic Finsler tori without conjugate points

Abstract: We show that analytic, k-basic Finsler metrics in the two torus without conjugate points are analytically integrable, in the sense that the unit tangent bundle of the metric admits an analytic foliation by invariant Lagrangian graphs. This result, combined with the fact that C1,L integrable k-basic Finsler metrics in the two torus have zero flag curvature (Barbosa-Ruggiero [19]) implies that analytic k-basic Finsler metrics in two tori without conjugate points are flat, a positive answer to the so-called Hopf conjecture for tori without conjugate points. Since there are well known examples of non flat tori without conjugate points (Busemann was the first to show such examples) the Hopf conjecture is not true if we drop the k-basic assumption. As for higher dimensional tori, a quite simple argument based on Schur’s Lemma shows that the only Finsler, k-basic (3 + m)-tori are the flat ones for every m ≥ 0.