SECTIONAL CURVATURE AND SYMMETRICAL FIRST INTEGRALS OF GEODESIC-FLOW - A GENERALIZATION OF BOCHNER,S. THEOREM

Let (M, g) be a Riemannian manifold. We prove that the space of symmet ric tensors invariant under the geodesic flow, is a Lie algebra which contains, as a subalgebra, the Lie algebra of Killing vector fields, a nd which also contains the space of parallel symmetric tensors as an A belian subalgebra. Morever, we give a Weitzenbock decomposition of som e Laplace-Beltrami operator on symmetric tensors and prove a vanishing theorem which generalizes a theorem dire to S. Bochner [2]. (C) Acade mie des Sciences/Elsevier, Paris.