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Generalized Ricci Solitons

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Abstract

We introduce a class of overdetermined systems of partial differential equations of finite type on (pseudo-) Riemannian manifolds that we call the generalized Ricci soliton equations. These equations depend on three real parameters. For special values of the parameters they specialize to various important classes of equations in differential geometry. Among them there are: the Ricci soliton equations, the vacuum near-horizon geometry equations in general relativity, special cases of Einstein–Weyl equations and their projective counterparts, equations for homotheties and Killing’s equation. We also prolong the generalized Ricci soliton equations and, by computing differential constraints, we find a number of necessary conditions for a (pseudo-) Riemannian manifold \((M, g)\) to locally admit non-trivial solutions to the generalized Ricci soliton equations in dimensions 2 and 3. The paper provides also a collection of explicit examples of generalized Ricci solitons in dimensions 2 and 3 (in some cases).

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Acknowledgments

Both authors would like to acknowledge Wojciech Kryński for organizing the workshop “Geometry of Projective Structures and Differential Equations” in Warsaw, where this work was initialized. We also wish to thank Piotr Chruściel, Jacek Jezierski and Paul Tod for helpful discussions. Special thanks are due to Gil Bor for his help in Sect. 7.4.1. This research was supported by the Polish National Science Center (NCN) via Grant DEC-2013/09/B/ST1/01799.

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Authors and Affiliations

  1. Centrum Fizyki Teoretycznej, Polska Akademia Nauk, Al. Lotników 32/46, 02-668, Warsaw, Poland

    Paweł  Nurowski

  2. Institut für Differentialgeometrie, Leibniz Universität Hannover, Welfengarten 1, 30167, Hannover, Germany

    Matthew Randall

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  1. Paweł  Nurowski
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  2. Matthew Randall
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Correspondence to Matthew Randall.

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Nurowski, P., Randall, M. Generalized Ricci Solitons. J Geom Anal 26, 1280–1345 (2016). https://doi.org/10.1007/s12220-015-9592-8

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  • DOI: https://doi.org/10.1007/s12220-015-9592-8

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