Showing papers on "Unit tangent bundle published in 1968"

Prolongations of $G$-structures to tangent bundles

Abstract: The purpose of the present paper is to study the prolongations of G-structures on a manifold M to its tangent bundle T(M), G being a Lie subgroup of GL(n,R) with n = dim M. Recently, K. Yano and S. Kobayashi [9] studied the prolongations of tensor fields on M to T(M) and they proposed the following question: Is it possible to associate with each G-structure on M a naturally induced G-structure on T(M), where G′ is a certain subgroup of GL(2n,R)? In this paper we give an answer to this question and we shall show that the prolongations of some special tensor fields by Yano-Kobayashi — for instance, the prolongations of almost complex structures — are derived naturally by our prolongations of the classical G-structures. On the other hand, S. Sasaki [5] studied a prolongation of Riemannian metrics on M to a Riemannian metric on T(M), while the prolongation of a (positive definite) Riemannian metric due to Yano-Kobayashi is always pseudo-Riemannian on T(M) but never Riemannian. We shall clarify the circumstances for this difference and give the reason why the one is positive definite Riemannian and the other is not.