Killing symmetry on the Finsler manifold

Abstract

Symmetry and conservation law are discussed on the Finsler manifold M. We adopt the point Finsler approach, where we consider the geometry on a point manifold M not on TM. Generalized vector fields are defined on oriented curves on M, and Finsler non-linear connections are considered on M, not on the tangent space TM. Killing vector fields K are defined as generalized vector fields as , and the Killing symmetry is also reformulated simply as by using the Killing 1-form and the spray operator defined by using the non-linear connection. is related to the generalization of Killing tensors on the Finsler manifold, and our ansatz of and give an analytical method of finding higher derivative conserved quantities, which may be called hidden conserved quantities. We show two examples: the Carter constant on Kerr spacetime and the Runge–Lentz vectors in Newtonian gravity.