On a new class of infinitesimal group actions on pseudo-Riemannian manifolds

Abstract

Using the Lie derivative of the metric we define a class of Lie algebras of vector fields by generalising the concept of Killing vectors. As a Lie algebra they define locally a group action on the pseudo-Riemannian manifold through exponentiation. The motivation behind studying these infinitesimal group actions is the investigation of $\mathcal{I}$-degenerate pseudo-Riemannian spaces, i.e., spaces having identical polynomial curvature invariants. In particular, we show that all the known examples of $\mathcal{I}$-degenerate pseudo-Riemannian spaces possess such vector fields.