Benenti Tensors: A useful tool in Projective Differential Geometry

Abstract

Two metrics are said to be projectively equivalent if they share the same geodesics (viewed as unparametrized curves). The degree of mobility of a metric g is the dimension of the space of the metrics projectively equivalent to g. For any pair of metrics (g, ḡ) on the same manifold one can construct a (1, 1)- tensor L(g, ḡ) called the Benenti tensor. In this paper we discuss some geometrical properties of Benenti tensors when (g, ḡ) are projectively equivalent, particularly in the case of degree of mobility equal to 2.

Keywords

• projective connections
• benenti tensors
• projectively equivalent metrics
• levi-civita metrics
• 53a20
• 53b10