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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Geodesics in metrical connections


Author: Richard S. Millman
Journal: Proc. Amer. Math. Soc. 30 (1971), 551-555
MSC: Primary 53.70
DOI: https://doi.org/10.1090/S0002-9939-1971-0282312-X
MathSciNet review: 0282312
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Abstract: To each connection on a Riemannian manifold we define a tensor called the Q-tensor. We prove that two metrical connections have the same geodesics if and only if their Q-tensors are equal. We then show that any manifold of dimension greater than two admits many metrical connections having the same geodesics; in particular, the Q-tensor is a strictly weaker invariant than the torsion.


References [Enhancements On Off] (What's this?)

  • [1] Richard L. Bishop and Richard J. Crittenden, Geometry of manifolds, Pure and Applied Mathematics, Vol. XV, Academic Press, New York-London, 1964. MR 0169148
  • [2] Elie Cartan, Sur les variétés à connexion affine, et la théorie de la relativité généralisée (deuxième partie), Ann. Sci. École Norm. Sup. (3) 42 (1925), 17–88 (French). MR 1509263
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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0282312-X
Keywords: Metrical connections, geodesics, Q-tensor
Article copyright: © Copyright 1971 American Mathematical Society