Symmetries in 4-dimensional Manifolds

Abstract

In this paper some algebraic and geometrical properties of symmetries (taken here as Lie algebras of smooth Killing vector fields) on a 4 dimensional manifold of arbitrary signature will be described. The discussion will include the theory of the distributions arising from such vector fields, their resulting orbit and isotropy structure and certain stability properties which these orbits may, or may not, possess. A link between the isotropies and the restrictions on the fundamental tensors of Ricci and Weyl (in terms of the subalgebras of the Lie algebras o(4), o(1,3) and o(2,2)) will be briefly discussed.