A x -operator on complete riemannian manifolds

Abstract

In this paper we give a generalisation of Kostant’s Theorem about theAx-operator associated to a Killing vector fieldX on a compact Riemannian manifold. Kostant proved (see [6], [5] or [7]) that in a compact Riemannian manifold, the (1, 1) skew-symmetric operatorAx=Lx−≡x associated to a Killing vector fieldX lies in the holonomy algebra at each point. We prove that in a complete non-compact Riemannian manifold (M, g) theAx-operator associated to a Killing vector field, with finite global norm, lies in the holonomy algebra at each point. Finally we give examples of Killing vector fields with infinite global norms on non-flat manifolds such thatAx does not lie in the holonomy algebra at any point.