On the projective geometry of sprays

摘要

After a careful study of the mixed curvatures of the Berwald-type (in particular, Berwald) connections, we present an axiomatic description of the so-called Yano-type Finsler connections. Using the Yano connection, we derive an intrinsic expression of Douglas' famous projective curvature tensor and we also represent it in terms of the Berwald connection. Utilizing a clever observation of Z. Shen, we show in a coordinate-free manner that a "spray manifold" is projectively equivalent to an affinely connected manifold iff its Douglas tensor vanishes. From this result we infer immediately that the vanishing of the Douglas tensor implies that the projective Weyl tensor of the Berwald connection "depends only on the position".