Four-dimensional almost Einstein manifolds with skew-circulant stuctures

Abstract

We consider a four-dimensional Riemannian manifold M with an additional structure S, whose fourth power is minus identity. In a local coordinate system the components of the metric g and the structure S form skew-circulant matrices. Both structures S and g are compatible, such that an isometry is induced in every tangent space of M. By a special identity for the curvature tensor, generated by the Riemannian connection of g, we determine classes of Einstein and almost Einstein manifolds. For such manifolds we obtain propositions for the sectional curvatures of some characteristic 2-planes in a tangent space of M. We consider a Hermitian manifold associated with the studied manifold and find conditions for g, under which it is a Kähler manifold. We construct some examples of the considered manifolds on Lie groups.

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Acknowledgements

The authors are grateful to Prof. G. Dzhelepov for his valuable comments.

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Correspondence to Iva Dokuzova.

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This work is partially supported by Project MU19-FMI-020 of the Scientific Research Fund, Paisii Hilendarski University of Plovdiv, Bulgaria.

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Dokuzova, I., Razpopov, D. Four-dimensional almost Einstein manifolds with skew-circulant stuctures. J. Geom. 111, 9 (2020). https://doi.org/10.1007/s00022-020-0521-z

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Keywords

  • Riemannian manifold
  • Einstein manifold
  • sectional curvatures
  • Ricci curvature
  • Lie group

Mathematics Subject Classification

  • 53B20
  • 53C15
  • 53C25
  • 53C55