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Geodesic equivalence via integrability


Topalov, P; Matveev, V (2003). Geodesic equivalence via integrability. Geometriae Dedicata, 96(1):91-115.

Abstract

We suggest a construction that, given an orbital diffeomorphism between two Hamiltonian systems, produces integrals of them. We treat geodesic equivalence of metrics as the main example of it. In this case, the integrals commute; they are functionally independent if the eigenvalues of the tensor g ig¯j are all different; if the eigenvalues are all different at least at one point then they are all different at almost each point and the geodesic flows of the metrics are Liouville integrable. This gives us topological obstacles to geodesic equivalence.

We suggest a construction that, given an orbital diffeomorphism between two Hamiltonian systems, produces integrals of them. We treat geodesic equivalence of metrics as the main example of it. In this case, the integrals commute; they are functionally independent if the eigenvalues of the tensor g ig¯j are all different; if the eigenvalues are all different at least at one point then they are all different at almost each point and the geodesic flows of the metrics are Liouville integrable. This gives us topological obstacles to geodesic equivalence.

Citations

24 citations in Web of Science®
27 citations in Scopus®
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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:geodesically equivalent metrics - integrable systems - Levi-Civita coordinates - projectively equivalent metrics
Language:English
Date:2003
Deposited On:29 Nov 2010 16:26
Last Modified:05 Apr 2016 13:25
Publisher:Springer
ISSN:0046-5755
Publisher DOI:https://doi.org/10.1023/A:1022166218282
Related URLs:http://www.ams.org/mathscinet-getitem?mr=1956835
http://www.zentralblatt-math.org/zbmath/search/?q=an%3A1017.37029

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