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.

Lie Sphere Geometry

Abstract: This chapter presents the method of moving frames in Lie sphere geometry. This involves a number of new ideas, beginning with the fact that some Lie sphere transformations are not diffeomorphisms of space S3, but rather of the unit tangent bundle of S3. This we identify with the set of pencils of oriented spheres in S3, which is identified with the set \( \varLambda \) of all lines in the quadric hypersurface \( Q \subset \mathbf{P}(\mathbf{R}^{4,2}) \). The set \( \varLambda \) is a five-dimensional subspace of the Grassmannian G(2, 6). The Lie sphere transformations are the projective transformations of P(R4, 2) that send Q to Q. This is a Lie group acting transitively on \( \varLambda \). The Lie sphere transformations taking points of S3 to points of S3 are exactly the Mobius transformations, which form a proper subgroup of the Lie sphere group. In particular, the isometry groups of the space forms are natural subgroups of the Lie sphere group. There is a contact structure on \( \varLambda \) invariant under the Lie sphere group. A surface immersed in a space form with a unit normal vector field has an equivariant Legendre lift into \( \varLambda \). A surface conformally immersed into Mobius space with an oriented tangent sphere map has an equivariant Legendre lift into \( \varLambda \). This chapter studies Legendre immersions of surfaces into this homogeneous space \( \varLambda \) under the action of the Lie sphere group. A major application is a proof that all Dupin immersions of surfaces in a space form are Lie sphere congruent to each other.

Killing vector fields on tangent bundles with Cheeger-Gromoll metric

Abstract: This paper deals with properties of the Cheeger-Gromoll metric g introduced in 1988 by Musso and Tricerri on the tangent bundle TM associated to a given Riemannian metric ðM; gÞ. One can find here essentially the two following results: 1. A classification of Killing vector fields on ðTM; gÞ 2. A generalization of a result of M. Sekizawa concerning the non rigidity of the Cheeger-Gromoll metric.

Relations between linear connections on the tangent bundle and connections on the jet bundle of a fibred manifold

Abstract: All natural operations transforming linear connections on the tangent bundle of a fibred manifold to connections on the 1-jet bundle are classified. It is proved that such operators form a 2-parameter family (with real coefficients).

A remark on bigness of the tangent bundle of a smooth projective variety and D-simplicity of its section rings

Abstract: We point out a connection between bigness of the tangent bundle of a smooth projective variety X over ℂ and simplicity of the section rings of X as modules over their rings of differential operators. As a consequence, we see that the tangent bundle of a smooth projective toric variety or a (partial) flag variety is big. Some other applications and related questions are discussed.