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manuscripta mathematica

, Volume 157, Issue 1–2, pp 279–294 | Cite as

The affine quasi-Einstein Equation for homogeneous surfaces

  • M. Brozos-Vázquez
  • E. García-RíoEmail author
  • P. Gilkey
  • X. Valle-Regueiro
Article

Abstract

We study the affine quasi-Einstein Equation for homogeneous surfaces. This gives rise through the modified Riemannian extension to new half conformally flat generalized quasi-Einstein neutral signature (2, 2) manifolds, to conformally Einstein manifolds and also to new Einstein manifolds through a warped product construction.

Mathematics Subject Classification

53C21 53B30 53C24 53C44 

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References

  1. 1.
    Afifi, Z.: Riemann extensions of affine connected spaces. Q. J. Math. Oxf. Ser. (2) 5, 312–320 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brozos-Vázquez, M., García-Río, E., Gilkey, P.: Homogeneous affine surfaces: killing vector fields and gradient Ricci solitons. J. Math. Soc. Japan 70, 1–45 (To appear) Google Scholar
  3. 3.
    Brozos-Vázquez, M., García-Río, E., Gilkey, P., Valle-Regueiro, X.: Half conformally flat generalized quasi-Einstein manifolds. arXiv:1702.06714
  4. 4.
    Brozos-Vázquez, M., García-Río, E., Gilkey, P., Valle-Regueiro, X.: A natural linear equation in affine geometry: the affine quasi-Einstein equation. arXiv:1705.08352
  5. 5.
    Calviño-Louzao, E., García-Río, E., Gilkey, P., Vázquez-Lorenzo, R.: The geometry of modified Riemannian extensions. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465, 2023–2040 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Derdzinski, A., Roter, W.: Walker’s theorem without coordinates. J. Math. Phys. 47(6), 062504 8 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Eisenhart,L. P.: Non-Riemannian geometry (Reprint of the 1927 original). Am. Math. Soc. Colloq. Publ. 8, American Mathematical Society, Providence, RI (1990)Google Scholar
  8. 8.
    Kim, D.-S., Kim, Y.H.: Compact Einstein warped product spaces with nonpositive scalar curvature. Proc. Am. Math. Soc. 131, 2573–2576 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kühnel, W., Rademacher, H.B.: Conformal transformations of pseudo-Riemannian manifolds. Recent developments in pseudo-Riemannian geometry. ESI Lectures in Mathematics and Physics, European Mathematical Society, Zürich, pp. 261–298 (2008)Google Scholar
  10. 10.
    Nomizu, K., Sasaki, T.: Affine Differential Geometry. Cambridge Tracts in Mathematics, vol. 111. Cambridge University Press, Cambridge (1994)zbMATHGoogle Scholar
  11. 11.
    Opozda, B.: A classification of locally homogeneous connections on 2-dimensional manifolds. Differ. Geom. Appl. 21, 173–198 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Patterson, E.M., Walker, A.G.: Riemann extensions. Q. J. Math. Oxf. Ser. (2) 3, 19–28 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Steglich, Ch.: Invariants of conformal and projective structures. Results Math. 27, 188–193 (1995)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Differential Geometry and its Applications Research Group, Escola Politécnica SuperiorUniversidade da CoruñaFerrolSpain
  2. 2.Faculty of MathematicsUniversity of Santiago de CompostelaSantiago de CompostelaSpain
  3. 3.Mathematics DepartmentUniversity of OregonEugeneUSA

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