Killing vector fields and holonomy algebras

Abstract

We prove that for each Killing vector field X X on a complete Riemannian manifold, whose orthogonal distribution is involutive, the ( 1 , 1 ) (1,1) skew-symmetric operator A X {A_X} associated to X X by A X = L X − ∇ X {A_X} = {L_X} - {\nabla _X} lies in the holonomy algebra at each point. By using the same techniques, we also study when that operator lies in the infinitesimal and local holonomy algebras respectively.