LINEAR CONNECTIONS IN THE BUNDLE OF LINEAR FRAMES

Abstract

Let L(M ) be the bundle of all linear frames over M , u an arbitrarily given point of L(M ), and ∇ : X(M ) × X(M ) → X(M ) a linear connection on M . Then the following results are well known: the horizontal subspace and the connection form at the point u may be written in terms of local coordinates of u ∈ L(M ) and Christoffel's symbols defined by ∇. These results are very fundamental on the study of the theory of connections. In this paper we show that the local expressions of those at the point u do not depend on the choice of a local coordinate system around the point u ∈ L(M ), which is rarely seen. Moreover we give full explanations for the following fact: the covariant derivative on M which is defined by the parallelism on L(M ), determined from the connection form above, coincides with ∇.

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