Projectively Equivalent Riemannian Spaces as Quasi-bi-Hamiltonian Systems

The class of Riemannian spaces admitting projectively, or geodesically, equivalent metrics is very closely related to a certain class of spaces for which the Hamilton-Jacobi equation for geodesics is separable. This fact is established, and its consequences explored, by showing that when a Riemannia...

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Bibliographic Details
Published inActa applicandae mathematicae Vol. 77; no. 3; p. 237
Main Author Crampin, M
Format Journal Article
LanguageEnglish
Published Dordrecht Springer Nature B.V 01.07.2003
Subjects
Integrals
Mathematical analysis
Studies
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Summary:The class of Riemannian spaces admitting projectively, or geodesically, equivalent metrics is very closely related to a certain class of spaces for which the Hamilton-Jacobi equation for geodesics is separable. This fact is established, and its consequences explored, by showing that when a Riemannian space has a projectively equivalent metric its geodesic flow is a quasi-bi-Hamiltonian system. The existence of involutive first integrals of the geodesic flow, quadratic in the momenta, follows by a standard type of argument. When these integrals are independent they generate a Stackel system. [PUBLICATION ABSTRACT]
ISSN:0167-8019
1572-9036
DOI:10.1023/A:1024907509042