  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.
