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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

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Splitting and gluing lemmas for geodesically equivalent pseudo-Riemannian metrics
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by Alexey V. Bolsinov and Vladimir S. Matveev PDF
Trans. Amer. Math. Soc. 363 (2011), 4081-4107 Request permission


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
  • MR Author ID: 248231
  • Email:
  • Vladimir S. Matveev
  • Affiliation: Institute of Mathematics, Friedrich-Schiller University Jena, 07737 Jena, Germany
  • MR Author ID: 609466
  • Email:
  • Received by editor(s): April 16, 2009
  • Published electronically: March 21, 2011
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 363 (2011), 4081-4107
  • MSC (2010): Primary 53A20, 53A35, 53A45, 53B20, 53B30, 53C12, 53C21, 53C22, 37J35
  • DOI:
  • MathSciNet review: 2792981