Geodesic
On conformal Killing symmetric tensor fields on Riemannian manifolds
Matematičeskie trudy, Tome 13 (2010) no. 1, pp. 85-145.

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A vector field on Riemannian manifold is called conformal Killing if it generates oneparameter group of conformal transformation. 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. In this paper, we prove that a trace-free conformal Killing tensor field is identically 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 manifold with boundary to a sum of three fields of special types. We also establish triviality of the space of trace-free conformal Killing tensor fields on some closed manifolds. 
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N. S. Dairbekov; V. A. Sharafutdinov. On conformal Killing symmetric tensor fields on Riemannian manifolds. Matematičeskie trudy, Tome 13 (2010) no. 1, pp. 85-145. http://geodesic.mathdoc.fr/item/MT_2010_13_1_a4/

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