Linear Connections on Tangent Bundles

The Tangent Bundle of a Locally Symmetric Space

“…It was shown in an earlier paper [1], that a linear connection V on a C"*-manifold M, determines a connection on the tangent bundle TM of M. In the present paper, a second connection is defined on TM, which appears to be more closely related bo the given connection on M, at least in the case of zero torsion, f In particular, with this connection, TM is locally symmetric if and only if M is locally symmetric.…”

“…a C™-tensor field of type (1, 1)) then I) is a vertical fields Proof. If (1) is satisfied by a vector field D, then by Lemma 1 of [1], D is unique. Also if D is an.…”

“…Also if D is an. endomorphism then D is the vertical vector field of Proposition 1 of [1]. Finally, for a general D, let X be the C™ -vector field on M arising from D; thus Xf= Df, for all / e FM.…”

“…Finally, for a general D, let X be the C™ -vector field on M arising from D; thus Xf= Df, for all / e FM. Let V be a torsion-free linear connection on M. Then D-Vx is an endomorphism and so D-Vx is a vector fieid satisfying (1). By Lemma 2 of [1] we have…”

“…We note that if D x , D 2 are derivations of degree zero, and if/ l3 f 2 eFM, then A A H = A A = ( / i OR eceived 12 May 1964. f We shall consider only linear connections. J Extensive use will be made of the methods and results of [1], and the reader who wishes to study the present paper in detail is advised to read [1] first of all.…”