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The General Solution of the Eisenhart Equation and Projective Motions of Pseudo-Riemannian Manifolds

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Abstract

The solution of the Eisenhart equation for pseudo-Riemannian manifolds (Mn,g) of arbitrary signature and any dimension is obtained. Thereby, pseudo-Riemannian h-spaces (i.e., spaces admitting nontrivial solutions h ≠ cg of the Eisenhart equation) of all possible types determined by the Segrè characteristic χ of the bilinear form h are found. Necessary and sufficient conditions for the existence of an infinitesimal projective transformation in (Mn,g) are given. The curvature 2-form of a (rigid) h-space of type χ = {r1, …, rk} is calculated and necessary and sufficient conditions for this space to have constant curvature are obtained.

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Funding

This work was supported by the National Science Foundation (NSF) under grant DMS-0901230.

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Authors and Affiliations

  1. Kazan (Volga Region) Federal University, Kazan, 420008, Russia

    A. V. Aminova

  2. Queens College, City University of New York (CUNY), Queens, New York, NY, 11367, USA

    M. N. Sabitova

Authors
  1. A. V. Aminova
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  2. M. N. Sabitova
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Corresponding authors

Correspondence to A. V. Aminova or M. N. Sabitova.

Additional information

Russian Text © The Author(s), 2020, published in Matematicheskie Zametki, 2020, Vol. 107, No. 6, pp. 803–816.

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Aminova, A.V., Sabitova, M.N. The General Solution of the Eisenhart Equation and Projective Motions of Pseudo-Riemannian Manifolds. Math Notes 107, 875–886 (2020). https://doi.org/10.1134/S0001434620050181

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  • DOI: https://doi.org/10.1134/S0001434620050181

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