# Commutative conservation laws for geodesic flows of metrics admitting projective symmetry

Topalov, P (2002). Commutative conservation laws for geodesic flows of metrics admitting projective symmetry. Mathematical Research Letters, 9(1):65-72.

## Abstract

We prove that the geodesic flow of a pseudo-Riemannian metric $g$ that admits a "nontrivial" projective symmetry $X$ is completely integrable. Nontriviality condition of the projective symmetry is expressed in the terms of the invariants of the pair forms $g$ and $L_Xg$, where $L_X$ denotes the Lie derivative with respect to the vector field $X$. The theorem we propose can be considered as a "commutative" analog of the Noether theorem.

We prove that the geodesic flow of a pseudo-Riemannian metric $g$ that admits a "nontrivial" projective symmetry $X$ is completely integrable. Nontriviality condition of the projective symmetry is expressed in the terms of the invariants of the pair forms $g$ and $L_Xg$, where $L_X$ denotes the Lie derivative with respect to the vector field $X$. The theorem we propose can be considered as a "commutative" analog of the Noether theorem.

## Citations

8 citations in Web of Science®
8 citations in Scopus®

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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics English 2002 29 Nov 2010 16:27 05 Apr 2016 13:25 International Press 1073-2780 First published in [Mathematical Research Letters] in [9 (2002), no. 1], published by International Press. Copyright © 2002 Mathematical Research Letters. All rights reserved. http://www.mrlonline.org/mrl/2002-009-001/2002-009-001-005.html http://www.ams.org/mathscinet-getitem?mr=1892314http://www.zentralblatt-math.org/zbmath/search/?q=an%3A05375219
Permanent URL: https://doi.org/10.5167/uzh-21971