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Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, Vol. 26, No. 2, pp. 397-413 (2010)
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Finsler-metrizabilities of spray manifolds

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Szabolcs Vattamány

College of Nyíregyháza

**Abstract:** A spray manifold (M,S) is said to be Finsler-metrizable in a broad sense or projectively Finsler, if there exists a Finsler structure L: TM->R such that the Finsler manifold (M,L) is projectively equivalent to (M,S). If, in particular, the canonical spray of (M,L) coincides with the given spray S, then we say that (M,S) is Finsler-metrizable in a natural sense or that S is a Finsler-variational spray. In his influential paper [3] M. Crampin presented a stimulating intrinsic reformulation of the famous Helmholtz conditions from the classical inverse problem of the calculus of variations through the existence of a 2-form on the tangent manifold. Prescribing some extra condition on this 2-form we derive necessary and sufficient conditions for metrizability of a spray in both senses.

**Keywords:** spray manifold, Finsler structure, projective change, metrizability

**Classification (MSC2000):** 53B40

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