Version 1
: Received: 10 June 2019 / Approved: 13 June 2019 / Online: 13 June 2019 (05:36:28 CEST)
Version 2
: Received: 27 May 2020 / Approved: 28 May 2020 / Online: 28 May 2020 (02:58:44 CEST)
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
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
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
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
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 K\"{a}hler manifold. We construct some examples of the considered manifolds on Lie groups.
Keywords
Riemannian manifold; Einstein manifold; sectional curvatures; Ricci curvature; Lie group
Subject
Computer Science and Mathematics, Geometry and Topology
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Received:
28 May 2020
Commenter:
Iva Dokuzova
Commenter's Conflict of Interests:
Author
Comment:
Theorem 8. was corrected. There was a technical errors. The proof of the Proposition 1 was corrected in order to be shorter. Corollary 1 was corrected.
Commenter: Iva Dokuzova
Commenter's Conflict of Interests: Author