On conformal Killing symmetric tensor fields on Riemannian manifolds

TL;DR

This paper studies trace-free conformal Killing symmetric tensor fields on Riemannian manifolds, proving their uniqueness under certain conditions and generalizing classical results about conformal Killing vector fields to higher-rank tensors.

Contribution

It establishes the vanishing of trace-free conformal Killing tensor fields when they vanish on a hypersurface and generalizes Bochner-Yano's theorem to higher-rank tensors under negative curvature.

Findings

Trace-free conformal Killing tensor fields are uniquely determined by their jets.

Such fields vanish if they vanish on some hypersurface.

No nontrivial trace-free conformal Killing tensors exist on negatively curved closed manifolds.

Abstract

A vector field on a Riemannian manifold is called conformal Killing if it generates one-parameter group of conformal transformations. The class of conformal Killing symmetric tensor fields of an arbitrary rank is a natural generalization of the class of conformal Killing vector fields, and appears in different geometric and physical problems. A symmetric tensor field is a trace-free field if the contraction of the field with the metric tensor is identically equal to zero. On a Riemannian manifold of dimension at least three, the space of trace-free conformal Killing symmetric tensor fields of arbitrary rank is of a finite dimension. On a two-dimensional manifold, the space can be of infinite dimension. Nevertheless, on a connected manifold of any dimension, a trace-free conformal Killing tensor field is uniquely determined by its $C^\infty$-jet at any point. We prove the statement: On a connected manifold, a trace-free conformal Killing tensor field is identically equal to zero if it vanishes on some hypersurface. This statement is a basis of the theorem on decomposition of a symmetric tensor field on a compact Riemannian manifold with boundary to a sum of three fields such that the first summand is a trace-free and divergence-free field, the second summand is a potential field with a trace-free potential vanishing on the boundary, and the last summand is a multiple of the metric tensor. The classical theorem by Bochner - Yano states the absence of conformal Killing vector fields on a closed manifold of negative Ricci curvature. We generalize the latter statements to arbitrary rank tensor fields but under the stronger hypothesis: the sectional curvature is assumed to be negative. There is no nontrivial trace-free conformal Killing symmetric tensor field of any rank on a closed negatively curved Riemannian manifold.