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Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, Vol. 26, No. 2, pp. 171-207 (2010)
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On the projective theory of sprays

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Sándor Bácsó and Zoltán Szilasi

University of Debrecen

**Abstract:** 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 Rapcsak equations.

**Keywords:** Ehresmann connections, sprays, Berwald curvature, affine curvature, Finsler functions, projective equivalence, metrizabilities of sprays, Rapcsák equations

**Classification (MSC2000):** 53C05; 53C60, 58G30

**Full text of the article:**

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