A NOTE ON SPRAYS OF ISOTROPIC CURVATURE

A basic goal of this paper is to calculate, Weyl curvature of R-flat (Ricci-flat) spray of isotropic curvature and a locally projectively R-flat (Ricci-flat) spray, which is a projective invariance. Besides, the equivalents of E ̅-curvature and H-curvature that are closely related to the mean Berwald curvature have been found for a locally projectively R-flat spray of isotropic curvature.

Inverse problem of sprays with scalar curvature

Every Finsler metric on a differential manifold induces a spray. The converse is not true. Therefore, it is one of the most fundamental problems in spray geometry to determine whether a spray is induced by a Finsler metric which is regular, but not necessary positive definite. This problem is called inverse problem. This paper discuss inverse problem of sprays with scalar curvature. In particular, we show that if such a spray [Formula: see text] on a manifold is of vanishing [Formula: see text]-curvature, but [Formula: see text] has not isotropic curvature, then [Formula: see text] is not induced by any (not necessary positive definite) Finsler metric. We also find infinitely many sprays on an open domain [Formula: see text] with scalar curvature and vanishing [Formula: see text]-curvature, but these sprays have no isotropic curvature. This contrasts sharply with the situation in Finsler geometry.

Sprays of isotropic curvature

In this paper, a new notion of isotropic curvature for sprays is introduced. We show that for a spray of scalar curvature, it is of isotropic curvature if and only if the non-Riemannian quantity [Formula: see text] vanishes. In fact, it is the first geometric quantity to show the spray of isotropic curvature even in the Finslerian case. How to determine a spray is induced by a Finsler metric or not is an interesting inverse problem. We study this problem when the spray is of isotropic curvature and show that a spray of zero curvature can be induced by a group of Finsler metrics. Further, an efficient way is given to construct a family of sprays of isotropic curvature which cannot be induced by any Finsler metric.