Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ALMOST EINSTEIN MANIFOLDS WITH CIRCULANT STRUCTURES

  • Dokuzova, Iva (Department of Algebra and Geometry University of Plovdiv "Paisii Hilendarski")
  • Received : 2016.08.06
  • Accepted : 2016.11.28
  • Published : 2017.09.01
Download PDF
⟨ Previous Next ⟩

Abstract

We consider a 3-dimensional Riemannian manifold M with a circulant metric g and a circulant structure q satisfying $q^3=id$. The structure q is compatible with g such that an isometry is induced in any tangent space of M. We introduce three classes of such manifolds. Two of them are determined by special properties of the curvature tensor. The third class is composed by manifolds whose structure q is parallel with respect to the Levi-Civita connection of g. We obtain some curvature properties of these manifolds (M, g, q) and give some explicit examples of such manifolds.

Keywords

References

  1. A. Borowiec, M. Ferraris, M. Francaviglia, and I. Volovich, Almost-complex and almost-product Einstein manifolds from a variational principle, J. Math. Phys. 40 (1999), no. 7, 3446-3464. https://doi.org/10.1063/1.532899
  2. F. M. Cabrera and A. Swann, Curvature of special almost Hermitian manifolds, Pacific J. Math. 228 (2006), no. 1, 165-184. https://doi.org/10.2140/pjm.2006.228.165
  3. E. Cotton, Sur les varietes a trois dimensions, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. (2) 1 (1899), no. 4, 385-438.
  4. I. Dokuzova, D. Razpopov, and G. Dzhelepov, Three-dimensional Riemannian manifolds with circulant structures, arXiv:1308.4834.
  5. G. Dzhelepov, I. Dokuzova, and D. Razpopov, On a three-dimensional Riemannian manifold with an additional structure, Plovdiv. Univ. Paisii Khilendarski Nauchn. Trud. Mat. 38 (2011), no. 3, 17-27.
  6. M. Falcitelli, A. Farinola, and S. Salamon, Almost-Hermitian geometry, Differential Geom. Appl. 4 (1994), no. 3, 259-282. https://doi.org/10.1016/0926-2245(94)00016-6
  7. A. Gray and L. Hervella, The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl. 123 (1980), 35-58. https://doi.org/10.1007/BF01796539
  8. R. M. Gray, Toeplitz and circulant matrices: A review, Found. Trends Commun. Inf. Theory 2 (2006), no. 3, 155-239. https://doi.org/10.1561/0100000006
  9. D. Mekerov, P-connection on Riemannian almost product manifolds, C. R. Acad. Bulgare Sci. 62 (2009), no. 11, 1363-1370.
  10. R. Mocanu and M. I. Munteanu, Gray curvature identities for almost contact metric manifolds, J. Korean Math. Soc. 47 (2010), no. 3, 505-521. https://doi.org/10.4134/JKMS.2010.47.3.505
  11. A. Naveira, A classification of Riemannian almost product manifolds, Rend. Mat. (7) 3 (1983), no. 3, 577-592.
  12. P. K. Rashevsky, Riemannian Geometry and Tensor Analysis, Nauka Eds, Moscow, 1967.
  13. K. Yano, Differential geometry on complex and almost complex spaces, International Series of Monographs in Pure and Applied Mathematics, Vol. 49, A Pergamon Press Book. The Macmillan Co., New York, 1965.