On the projective theory of sprays. (English) Zbl 1240.53047

Bácsó, Sándor; Szilasi, Zoltán

Acta Math. Acad. Paedagog. Nyházi. (N.S.) 26, No. 2, 171-207 (2010).

Summary: The paper aims to give a fairly self-contained survey on the fundamentals and the basic techniques of spray geometry, using a rigorously index-free formalism in the pull-back bundle framework, with applications to Finslerian sprays and metrizability problems. Thus we review a number of classically well-known facts from a modern viewpoint, and prove also known results using new ideas and tools. Among others, Laugwitz’s metrization theorem and the proof of the vanishing of the direction independent Landsberg and stretch tensor belong to this category. We present also some results we believe are new. We mention from this group the description of the projective factors which yield the invariance of the Berwald curvature under a projective change and the sufficient conditions of the Finsler metrizability of a spray in a broad sense deduced from the Rapcsák equations.

MSC:

53C05	Connections (general theory)
53C60	Global differential geometry of Finsler spaces and generalizations (areal metrics)

Keywords:

Ehresmann connections; sprays; Berwald curvature; affine curvature; Finsler functions; projective equivalence; metrizabilities of sprays, Rapcsk equations