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Journal of the London Mathematical Society Advance Access originally published online on July 16, 2009
Journal of the London Mathematical Society 2009 80(2):341-356; doi:10.1112/jlms/jdp032
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© 2009 London Mathematical Society

A Fubini theorem for pseudo-Riemannian geodesically equivalent metrics

Alexey V. Bolsinov

School of Mathematics
Loughborough University
Loughborough
LE11 3TU
United Kingdom

Volodymyr Kiosak

Institute of Physics and Mathematics
K. Ushynsky South Ukraine Pedagogical University
Odessa
Ukraine
vkiosak@ukr.net

Vladimir S. Matveev

Institute of Mathematics
Friedrich-Schiller-Universität Jena
07737 Jena
Germany
vladimir.matveev@uni-jena.de

We generalize the following classical result of Fubini to pseudo-Riemannian metrics: if three essentially different metrics on an (n ≥ 3)-dimensional manifold M share the same unparametrized geodesics, and two of them (say, g and g) are strictly nonproportional (that is, the minimal polynomial of the g-self-adjoint (1, 1)-tensor defined by g coincides with the characteristic polynomial) at least at one point, then they have constant sectional curvature.

2000 Mathematics Subject Classification 53A20, 53B21, 53C22, 53C50, 53D25, 70G45, 70H06, 70H33, 58J60.

Received August 25, 2008; revised March 17, 2009; published online July 16, 2009.


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