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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Splitting and gluing lemmas for geodesically equivalent pseudo-Riemannian metrics


Authors: Alexey V. Bolsinov and Vladimir S. Matveev
Journal: Trans. Amer. Math. Soc. 363 (2011), 4081-4107
MSC (2010): Primary 53A20, 53A35, 53A45, 53B20, 53B30, 53C12, 53C21, 53C22, 37J35
Published electronically: March 21, 2011
MathSciNet review: 2792981
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Abstract: Two metrics $ g $ and $ \bar g$ are geodesically equivalent if they share the same (unparameterized) geodesics. We introduce two constructions that allow one to reduce many natural problems related to geodesically equivalent metrics, such as the classification of local normal forms and the Lie problem (the description of projective vector fields), to the case when the $ (1,1)-$tensor $ G^i_j:= g^{ik}\bar g_{kj}$ has one real eigenvalue, or two complex conjugate eigenvalues, and give first applications. As a part of the proof of the main result, we generalise the Topalov-Sinjukov (hierarchy) Theorem for pseudo-Riemannian metrics.


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Additional Information

Alexey V. Bolsinov
Affiliation: School of Mathematics, Loughborough University, Loughborough, LE11 3TU, United Kingdom
Email: A.Bolsinov@lboro.ac.uk

Vladimir S. Matveev
Affiliation: Institute of Mathematics, Friedrich-Schiller University Jena, 07737 Jena, Germany
Email: vladimir.matveev@uni-jena.de

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05187-1
PII: S 0002-9947(2011)05187-1
Received by editor(s): April 16, 2009
Published electronically: March 21, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.