A generalization of Weyl's theorem on projectively related affine connections |
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Authors: | Jó zsef Szilasi |
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Affiliation: | Institute of Mathematics, University of Debrecen, H-4010 Debrecen, P.O.B. 12, Hungary |
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Abstract: | From a physical point of view, the geodesics in a four-dimensional Lorentzian spacetime are really significant only as point sets. In 1921 Weyl proved that two torsion-free covariant derivative operators DM and on a manifold M have the same geodesics with possibly different parametrizations if and only if there is a 1-form α on M such that , where 1 is the identity (1,1) tensor on M. By a theorem of Ambrose, Palais and Singer [1], torsion-free covariant derivative operators are generated by affine sprays, and vice versa. More generally, any (not necessarily affine) spray induces a number of covariant derivatives in the tangent bundle τ of M, or in the pull-back bundle τ∗τ. We show that in the context of sprays, similarly to Weyl's relation, a correspondence between the Yano derivatives can be detected. |
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Keywords: | 53C22 53C80 |
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