HORIZONTAL SUBSPACES IN THE BUNDLE OF LINEAR FRAMES

Abstract

Let L(M) be the bundle of all linear frames over a smooth manifold M, <TEX>$u$</TEX> an arbitrarily given point of L(M), and <TEX>${\nabla}:\mathfrak{X}(M){\times}\mathfrak{X}(M){\rightarrow}\mathfrak{X}(M)$</TEX> a linear connection on M. Then the following result is well known: the horizontal subspace at the point <TEX>$u$</TEX> may be written in terms of local coordinates of <TEX>$u{\in}L(M)$</TEX> and Christoel's symbols defined by <TEX>${\nabla}$</TEX>. This result is very fundamental on the study of the theory of connections. In this paper we show that the local expression of the horizontal subspace at the point u does not depend on the choice of a local coordinate system around the point <TEX>$u{\in}L(M)$</TEX>, which is rarely seen.