Vanishing theorems for higher-order Killing and Codazzi

Abstract

A Killing p-tensor (for an arbitrary natural number p ≥ 2) is a sym­metric p-tensor with vanishing symmetrized covariant derivative. On the other hand, Codazzi p-tensor is a symmetric p-tensor with symmetric co­variant derivative. Let M be a complete and simply connected Riemanni­an manifold of nonpositive (resp. non-negative) sectional curvature. In the first case we prove that an arbitrary symmetric traceless Killing p-tensor is parallel on M if its norm is a -function for some q > 0. If in addition the volume of this manifold is infinite, then this tensor is equal to zero. In the second case we prove that an arbitrary traceless Codazzi p-tensor is equal to zero on a noncompact manifold M if its norm is a -function for some .

Keywords

• complete riemannian manifold
• killing and codazzi ten­sors
• vanishing theorem