Hessian Geometry on Lagrange Spaces

We extend the correspondence between Hessian and K？hler metrics and curvatures to Lagrange spaces. 1. Introduction Hessian geometry on locally affine manifolds was studied by several authors, particularly, Cheng and Yau [1] and Shima [2]. Shima introduced a notion of Hessian curvature, which is a finer invariant than Riemannian curvature (the Riemannian curvature of metrics defined by the Hessian of a function was studied extensively, for example, [3]) and is related to the curvature of an associated K？hler metric on the tangent manifold (the total space of the tangent bundle). A Lagrange space is a manifold with a regular Lagrangian, also called a Lagrange metric, on its tangent manifold [4]. The latter has the vertical foliation by fibers and the fiber-wise Hessian of the Lagrangian defines (pseudo)Hessian (“pseudo” is added if the metric is not positive definite) metrics of the fibers. In this note, we extend the correspondence Hessian versus K？hler to the vertical foliation of the tangent manifold of a Lagrange space (Section 3). The subject of the note is not Lagrangian dynamics but Hessian geometry and curvature in the context of Lagrange spaces, which are a generalization of (pseudo)Finsler spaces. The study of curvature is motivated by the general principle that curvature invariants differentiate between spaces of a given type. We will begin by recalling the basics of Hessian and tangent bundle geometry (Section 2) (since the reader is not supposed to be an expert on any of these) and by some required preparations. In an appendix we give index-free proofs of some properties of Hessian curvature established via local coordinates in [2]. We work in the category and use the standard notation of differential geometry [5]. 2. Preliminaries This is a preliminary section where we recall Hessian metrics and curvature and the basics of the geometry of tangent bundles. We refer to [2] for Hessian geometry and to [4, 6] for the tangent bundle geometry. 2.1. Hessian Geometry Let be a locally affine manifold with the flat, torsionless connection . A (pseudo)Hessian metric (structure) on is a (pseudo)Riemannian metric such that where is an open covering of , are local, parallel vector fields, and . If (1) holds on with a function , the metric is globally (pseudo)Hessian. Since local parallel vector fields are of the form (in the paper we use the Einstein summation convention), where are local affine coordinates and ., (1) is equivalent to Let be an arbitrary (pseudo)Riemannian metric on . The formula defines a tensor, which we call the Cartan tensor. If the