On 3-dimensional almost Einstein manifolds with circulant structures

A 3-dimensional Riemannian manifold equipped with a tensor structure of type 1,1 , whose third power is the identity, is considered. This structure and the metric have circulant matrices with respect to some basis, i.e. these structures are circulant. An associated manifold, whose metric is expressed by both structures, is studied. Three classes of such manifolds are considered. Two of them are determined by special properties of the curvature tensor of the manifold. The third class is composed by manifolds whose structure is parallel with respect to the Levi-Civitaconnection of the metric. Some geometric characteristics of these manifolds are obtained. Examples of such manifolds are given

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  • [1] Dajczer M, Nomizu K. On sectional curvature of indefinite metrics. II Mathematische Annalen 1980; 247 (3): 279-282. doi: 10.1007/BF01348960
  • [2] Di Scala AJ, Vezzoni L. Gray identities, canonical connections and integrability. Proceedings of the Edinburgh Mathematical Society 2010; 53 (3): 657-674. doi: 10.1017/S0013091509000157
  • [3] Dokuzova I. Almost Einstein manifolds with circulant structures. Journal of the Korean Mathematical Society 2017; 54 (5): 1441-1456. doi: 10.4134/JKMS.j160524 [4] Dokuzova I. On a Riemannian manifolds with a circulant structure whose third power is the identity. Filomat 2018; 32 (10): 3529-3539. doi: 10.2298/FIL1810529D
  • [5] Dzhelepov G, Dokuzova I, Razpopov D. On a three-dimensional Riemannian manifold with an additional structure. Plovdiv University Paissii Hilendarski Scientific Works - Mathematics 2011; 38 (3): 17-27.
  • [6] Dzhelepov G, Razpopov D, Dokuzova I. Almost conformal transformation in a class of Riemannian manifolds. In: Procceedings of the Anniversary International Conference – REMIA; Plovdiv, Bulgaria; 2010. pp. 125-128.
  • [7] Dzhelepov G. Spheres and circles in the tangent space at a point on a Riemannian manifold with respect to an indefinite metric. Novi Sad Journal of Mathematics 2018; 48 (1): 143-150. doi: 10.130755/NSJOM.07233
  • [8] Gray A, Hervella LM. The sixteen classes of almost Hermitian manifolds and their linear invariants. Annali di Matematica Pura ed Applicata 1980; 123 (4) : 35-58. doi: 10.1007/BF01796539
  • [9] Gray A. Curvature identities for Hermitian and almost Hermitian manifolds. Tôhoku Mathematical Journal 1976; 28 (4): 601-612. doi: 10.2748/tmj/1178240746
  • [10] Gribacheva D, Mekerov D. Conformal Riemannian P -manifolds with connections whose curvature tensors are Riemannan P -tensors. Journal of Geometry 2014; 105 (2): 273-286. doi: 10.1007/s00022-013-0206-y
  • [11] Mekerov D. On Riemannian almost product manifolds with nonintegrable structure. Journal of Geometry 2008; 89 (1-2): 119-129. doi: 10.1007/s00022-008-2084-9
  • [12] Naveira AM. A classification of Riemannian almost product manifolds. Rendiconti di Matematica e delle sue Applicazioni 1983; 3 (3): 577-592.
  • [13] Prvanović M. Conformally invariant tensors of an almost Hermitian manifold associated with the holomorphic curvature tensor. Journal of Geometry 2012; 103 (1): 89-101. doi: 10.1007/s00022-012-0111-9
  • [14] Staikova M, Gribachev K. Canonical connections and their conformal invariants on Riemannian almost productmanifolds. Serdica 1992; 18 (3-4): 150-161.
  • [15] Vanhecke L. Some almost Hermitian manifolds with constant holomorphic sectional curvature. Journal of Differential Geometry 1977; 12 (4): 461-471. doi: 10.4310/jdg/1214434217
  • [16] Yano K. Differential geometry on complex and almost complex spaces. International Series of Monographs in Pure and Applied Mathematics 49. New York, NY, USA: A Pergamon Press Book The Macmillan and Co., 1965.
  • [17] Yu C. Curvature identities on almost Hermitian manifolds and application. Science China Mathematics 2017; 60 (2): 285-300. doi: 10.1007/s11425-016-0022-6