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The *-Ricci Tensor of Real Hypersurfaces in Symmetric Spaces of Rank One or Two

  • George Kaimakamis
  • Konstantina Panagiotidou
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 106)

Abstract

Complex projective and hyperbolic spaces, i.e. non-flat complex space forms, are symmetric spaces of rank one. Complex two-plane Grassmannians are symmetric spaces of rank two. Let M be a real hypersurface in a symmetric space of rank one or two. Many geometers, such as Berndt, Jeong, Kim, Ortega, Pérez, Santos, Suh, Takagi and others have studied real hypersurfaces in above spaces in terms of their operators and tensor fields. This paper will be divided into two parts. Firstly, results concerning real hypersurfaces in non-flat complex space forms in terms of their-Ricci tensor, S , which in case of real hypersurfaces was first studied by Hamada (Real hypersurfaces of complex space forms in terms of Ricci *-tensor. Tokyo J. Math. 25, 473–483 (2002)), will be presented. More precisely, it will be answered if there exist or not real hypersurfaces, whose-Ricci tensor is parallel, semi-parallel, i.e. R ⋅ S  = 0, or pseudo-parallel, i.e. \(R(X,Y ) \cdot S^{{\ast}} = L\{(X \wedge Y ) \cdot S^{{\ast}}\}\) with L ≠ 0 (Kaimakamis and Panagiotidou, Parallel-Ricci tensor of real hypersurfaces in \(\mathbb{C}P^{2}\) and \(\mathbb{C}H^{2}\). Taiwan. J. Math., to appear, DOI 10.11650/tjm.18.2014.4271; Kaimakamis and Panagiotidou, Conditions of parallelism of-Ricci tensor of real hypersurfaces in \(\mathbb{C}P^{2}\) and \(\mathbb{C}H^{2}\). Preprint). Secondly, the formula of-Ricci tensor of real hypersurfaces in complex two-plane Grassmannians will be provided (Panagiotidou, The-Ricci tensor of real hypersurfaces in complex two-plane Grassmannians, work in progress).

Keywords

Symmetric Space Ricci Tensor Real Hypersurface Shape Operator Complex Projective Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Japan 2014

Authors and Affiliations

  1. 1.Faculty of Mathematics and Engineering SciencesHellenic Military AcademyVari, AttikiGreece
  2. 2.Faculty of EngineeringAristotle University of ThessalonikiThessalonikiGreece

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