On conformal Killing symmetric tensor fields on Riemannian manifolds

Abstract

Abstract. A vector field on a Riemannian manifold is called conformal Killing if it generates one-parameter 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. We prove the statement: 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.

The X-Ray Transform for Connections in Negative Curvature

“…The situation is quite simple on manifolds with boundary: any twisted CKT that vanishes on part of the boundary must be identically zero. The next theorem extends [DS11] which considered the case of a trivial line bundle with flat connection. This result will be used as a component in the proof of Theorem 1.1 (for Γ = ∂M and π −1 Γ = ∂(SM )).…”

“…The dimension of Ker(X + | Ωm ) is a conformal invariant (see Section 3). In the case of the trivial line bundle with flat connection, twisted CKTs coincide with the usual CKTs, and these cannot exist on any manifold whose conformal class contains a metric with negative sectional curvature or a rank one metric with nonpositive sectional curvature [DS11,PSU14d].…”

“…To begin, we need to recall certain notions related to the geometry of the unit sphere bundle. We follow the setup and notation of [PSU14d]; for other approaches and background information see [GK80b,Sh94,Pa99,Kn02,DS11].…”

Some differential operators in the symmetric bundle

“…The Killing, conformal Killing and trace free conformal Killing tensor, that constitute a subclass of the class of symmetric tensors considered in our paper, have an application in various problems of geometry, physics and tomography (see eg. [3,12,13]). …”

On the perturbation theory in spatially closed background

“…Equation (9) is also a theorem in Riemannian geometry [19,20]. According to this theorem, every rank 2 symmetric tensor on a compact Riemannian manifold can be uniquely represented in form of Eq.…”