LINEAR CONNECTIONS IN THE BUNDLE OF LINEAR FRAMES

Let L(M) be the bundle of all linear frames over M, u an arbitrarily given point of L(M), and r : X(M) £ X(M) ! X(M) a linear connection on M. Then the follow-ing results are well known: the horizontal subspace and the connection form at the point u may be written in terms of lo-cal coordinates of u 2 L(M) and Christo？？el s symbols de¯ned by r. These results are very fundamental on the study of the theory of connections. In this paper we show that the local ex-pressions of those at the point u do not depend on the choice ofa local coordinate system around the point u 2 L(M), which is rarely seen. Moreover we give full explanations for the fol- lowing fact: the covariant derivative on M which is de¯ned by the parallelism on L(M), determined from the connection form above, coincides with r.