Classification of second orders symmetric tensors on manifolds through an associated fourth order tensor

Abstract

For a manifold M admitting a metric 1 and given a second order symmetric tensor T on M one can construct from 1 and (the trace-free part of) T a fourth order tensor E on M which is related in a one-to-one way with T and from which T may be readily obtained algebraically. In the case when dimM = 4 this leads to an interesting relationship between the Jordan-Segre algebraic classification of T, viewed as a linear map on the tangent space to M with respect to 1, and the Jordan-Segre classification of E, viewed as a linear map on the 6?dimensional vector space of 2?forms to itself (with respect to the usual metric on 2?forms). This paper explores this relationship for each of the three possible signatures for 1.