On Harmonic and Killing Vector Fields

Abstract

Publisher Summary This chapter discusses a formula that gives immediately the proofs of Bochner Theorems for an orientable space and enables to see clearly, the way the contrast between harmonic and Killing vector fields arises. From this general formula, a theorem can be deduced that states that, in a compact orientable Riemannian space with positive definite metric, if the space has negative Ricci curvature throughout, then there exists no continuous group of conformal transformations other than the identity. If the space admits a one-parameter group of motions, then the Lie derivative of any harmonic tensor with respect to this motion vanishes. In a compact orientable Riemannian space with positive definitemetric, one-parameter group of affine collineations must be that of motions.