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The structure of the Ricci tensor on locally homogeneous Lorentzian gradient Ricci solitons

Part of: Global differential geometry Local differential geometry

Published online by Cambridge University Press:  22 February 2018

M. Brozos-Vázquez ,
E. García-Río ,
S. Gavino-Fernández  and
P. Gilkey
M. Brozos-Vázquez
Affiliation:
Universidade da Coruña, Differential Geometry and its Applications Research Group, Escola Politécnica Superior, 15403 Ferrol, Spain (miguel.brozos.vazquez@udc.gal)
E. García-Río
Affiliation:
Faculty of Mathematics, University of Santiago de Compostela, 15782 Santiago de Compostela, Spain (eduardo.garcia.rio@usc.es; sandra.gavino@usc.es)
S. Gavino-Fernández
Affiliation:
Faculty of Mathematics, University of Santiago de Compostela, 15782 Santiago de Compostela, Spain (eduardo.garcia.rio@usc.es; sandra.gavino@usc.es)
P. Gilkey
Affiliation:
Mathematics Department, University of Oregon, Eugene, OR 97403, USA (gilkey@uoregon.edu)
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Abstract

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We describe the structure of the Ricci tensor on a locally homogeneous Lorentzian gradient Ricci soliton. In the non-steady case, we show that the soliton is rigid in dimensions 3 and 4. In the steady case we give a complete classification in dimension 3.

Keywords

locally homogeneous Lorentzian manifoldCahen–Wallach spacegradient Ricci solitonharmonic Weyl tensorWalker manifoldKilling vector field

MSC classification

Secondary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions 53B30: Lorentz metrics, indefinite metrics 53C24: Rigidity results 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 148 , Issue 3 , June 2018 , pp. 461 - 482
Copyright
Copyright © Royal Society of Edinburgh 2018