On the projective geometry of sprays
A manifold endowed with a spray \(S\) is called a spray manifold (path space, or -- classically -- path geometry). Two path geometries \((M, S_1)\) and \((M,S_2)\) are projective1y equivalent, if the geodesics of \(S_1\) and \(S_2\) are the same as point sets, i.e., if they differ only in a (strictly increasing) parameter transformation. Then \(S_2 = S_1 + \lambda C\), where \(C\) is the Liouville vector field and \(\lambda\) is a 1-homogeneous function. Conversely, any transition \(S \rightarrow S + \lambda C\) (with 1-homogeneous \(\lambda\)) is a projective change of the spray \(S\). In a path geometry there are two fundamental projectively invariant curvature tensors: the projective Weyl tensor and the projective Douglas tensor. The main purpose of the paper is to present a coordinate-free (and in this sense ``intrinsic'') construction of the Douglas tensor by means of the Frölicher-Nijenhuis theory of derivations. The program is carried out using the so-called Yano connection, whose axiomatic description and intrinsic construction from the given spray is also a novelty of the paper. As an application it is proved, using Z. Shen's ideas, that a ``volume spray manifold'' is projectively equivalent to an affinely connected manifold if its Douglas tensor vanishes. This is a global version of a celebrated result of J. Douglas. Then it is concluded that the vanishing of the Douglas tensor also implies that the projective Weyl tensor is a vertical lift (and so it ``does not depend on the position''; cf. \textit{S.~Bácsó} and \textit{M.~Matsumoto} [Publ. Math. 53, No. 3-4, 423-438 (1998; Zbl 0924.53018)]). Reviewer's remark: In the course of printing a fatal misprint appeared (instead of \(\chi \) the symbol 10 everywhere), so the paper has been reprinted.
scientific article; zbMATH DE number 1485494