Chapter I Linear connections on a space of linear elements

Abstract

(abstract) Let M be a differentiable manifold of dimension n of class C∞. Let p: V(M) → M be the fibre bundle of non-zero tangent vectors to M with fibre type Rn − {0} with structure group GL(n, R) the general linear group in n real variables. We denote by π: W(M) → M the fibre bundle of oriented directions tangent to M. Let E(M) be the linear fibre bundle of frames on M and p − 1 E(M) the induced fibre bundle of E(M) by p. An infinitesimal connection on p − 1 E(M) is called a linear connection of vectors ([1]). The study of this connection leads us to single out a condition of regularity (§5). In this case, independent tensor forms can be introduced on V(M). To a regular linear connection of vectors are associated canonically two torsion tensors S T as well as three curvature tensors R P and Q; we find expressions for them in (§7). In view of obtaining the formulas of the habitual linear connections we establish a reduction theorem (§8). With the help of covariant derivations of two types ∇ and ∇ • we form three Ricci identities for a vector field in the large sense (§9). In §10 we show that there exist between the two torsion tensors S T as well as among the three curvature tensors R P and Q of a general regular connection five identities called Bianchi identities. We then give explicit formulas for them.