The index of a geodesic in a Randers space and some remarks about the lack of regularity of the energy functional of a Finsler metric

Abstract: In a series of papers ([2, 3, 4]) the relations existing between the metric properties of Randers spaces and the conformal geometry of stationary Lorentzian manifolds were discovered and investigated. These relations were called in [4] \emph{Stationary-to-Randers Correspondence (SRC)}. In this paper we focus on one aspect of SRC, the equality between the index of a geodesic in a Randers space and that of its lightlike lift in the associated conformal stationary spacetime. Moreover we make some remarks about regularity of the energy functional of a Finsler metric on the infinite dimensional manifold of $H^1$ curves connecting two points, in connection with infinite dimensional techniques in Morse Theory.

Keywords: Stationary Lorentzian manifolds, lightlike geodesics, Morse index, Finsler metric, Randers space

Classification (MSC2000): 53C22; 53C50, 53C60, 58B20

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