A COMPLETE SET OF BIANCHI IDENTITIES FOR TENSOR-FIELDS ALONG THE TANGENT BUNDLE PROJECTION

The various derivations defined along the tangent bundle projection r in a series of papers by Martinez, Carinena and Sarlet are expressed a s components of a single linear connection del on E, the tangent bundl e of the evolution space E = R x TM. This connection is equivalent to a system of second-order ordinary differential equations (SODE) on M. Using the linear connection, we calculate the torsion and curvature of (E, del), the components of which are expressed in terms of the tenso rs along r defined by Martinez et al. From these, the full set of Bian chi identities are calculated. We also show that the generalized Jacob i equation, defined by several authors, is precisely the horizontal co mponent of the conventional Jacobi equation along geodesics of (E, del ). Finally, we use this to show that if a Jacobi field of the lift of a SODE solution is a certain lift, then it can be extended to a symmet ry of the SODE.