A complete set of Bianchi identities for tensor fields along the tangent bundle projection

Abstract

The various derivations defined along the tangent bundle projection tau in a series of papers by Martinez, Carinena and Sarlet (1992) are expressed as components of a single linear connection Del on E, the tangent bundle of the evolution space E=R*T M. 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 tensors along tau defined by Martinez et. al. From these, the full set of Bianchi identities are calculated. We also show that the generalized Jacobi equation, defined by several authors, is precisely the horizontal component 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 symmetry of the SODE.