Killing Tensors and Conformal Killing Tensors from Conformal Killing Vectors

Abstract

Koutras has proposed some methods to construct reducible proper conformal Killing tensors and Killing tensors (which are, in general, irreducible) when a pair of orthogonal conformal Killing vectors exist in a given space. We give the completely general result demonstrating that this severe restriction of orthogonality is unnecessary. In addition we correct and extend some results concerning Killing tensors constructed from a single conformal Killing vector. A number of examples demonstrate how it is possible to construct a much larger class of reducible proper conformal Killing tensors and Killing tensors than permitted by the Koutras algorithms. In particular, by showing that all conformal Killing tensors are reducible in conformally flat spaces, we have a method of constructing all conformal Killing tensors (including all the Killing tensors which will in general be irreducible) of conformally flat spaces using their conformal Killing vectors.

18 pages References added. Comments and reference to 2-dim case. Typos corrected

References in corpus (1)

• The non-existence of a Lanczos potential for the Weyl curvature tensor in dimensions n>=7

Cited by in corpus (25)

• Generalized Robertson-Walker space times, a survey
• Lie - point and variational symmetries in minisuperspace Einstein's gravity
• Black holes in N=8 supergravity from SO(4,4) hidden symmetries
• Perfect-fluid, generalised Robertson-Walker space-times, and Gray's decomposition
• $f(R)$-gravity from Killing Tensors
• Conformal Yano-Killing tensor for the Kerr metric and conserved quantities
• A note on Harada's Conformal Killing gravity
• On the geometry of Killing and conformal tensors
• Notes on Super Killing Tensors
• Variational Contact Symmetries of Constraint Lagrangians
• Slice-reducible conformal Killing tensors, photon surfaces and shadows
• Quadratic first integrals of autonomous conservative dynamical systems
• First integrals of holonomic systems without Noether symmetries
• Friedmann equations in the Codazzi parametrization of Cotton and extended theories of gravity and the Dark Sector
• Killing tensors in pp-wave spacetimes
• Conformal Killing gravity in static spherically-symmetric spacetimes
• Conformal Killing cosmology -- Geometry, dark sector, growth of structures, and big rip
• Integrable and Superintegrable Potentials of 2d Autonomous Conservative Dynamical Systems
• Invariant classification of the rotationally symmetric R-separable webs for the Laplace equation in Euclidean space
• Existence of gradient CKV and gradient conformally stationary LRS spacetimes
• Quadratic Killing structure Jacobi operator for real hypersurfaces in complex two-plane Grassmannians
• Quadratic first integrals of constrained autonomous conservative dynamical systems with fixed energy
• Killing-Yano tensors and surface terms
• Constraints on cosmological parameters and CMB first acoustic peak in conformal Killing gravity
• Gradient conformal stationarity and the CMC condition in LRS spacetimes