In this paper, we study the solvmanifolds constructed from any parabolic subalgebras of any semisimple Lie algebras. These solvmanifolds are naturally homogeneous submanifolds of symmetric spaces of noncompact type. We show that the Ricci curvatures of our solvmanifolds coincide with the restrictions of the Ricci curvatures of the ambient symmetric spaces. Consequently, all of our solvmanifolds...

We give an elementary construction of symplectic connections through reduction. This provides an elegant description of a class of symmetric spaces and gives examples of symplectic connections with Ricci type curvature, which are not locally symmetric; the existence of such symplectic connections was unknown. Key-words: Marsden-Weinstein reduction, symplectic connections, symmetric spaces MSC 2...

We study the ∗-Ricci operator on Hopf real hypersurfaces in complex quadric. prove that for quadric, tensor is symmetric if and only unit normal vector field singular. In following, we obtain of quadric symmetric, then both Reeb-flow-invariant Reeb-parallel. As correspondence to semi-symmetric Ricci tensor, give a classification with tensor.

Copyright q 2010 Zisheng Hu et al. 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. We extend the classical Bishop-Gromov volume comparison from constant Ricci curvature lower bound to radially symmetric Ricci curvature lower bound...

A Riemannian manifold (M, g) is semi-symmetric if (R(X,Y ) ◦ R)(U, V,W ) = 0. It is called pseudo-symmetric if R ◦ R = F, F being a given function of X, . . . ,W and g. It is called partially pseudosymmetric if this last relation is fulfilled by not all values of X, . . . ,W . Such manifolds were investigated by several mathematicians: I.Z. Szabó, S. Tanno, K. Nomizu, R. Deszcz and others. In t...

We describe and construct here pseudo-Hermitian structures θ without torsion (i.e. with transversal symmetry) whose Webster-Ricci curvature tensor is a constant multiple of the exterior differential dθ. We call these structures pseudo-Hermitian Einstein and our result states that they all can be derived locally from Kähler-Einstein metrics. Moreover, we discuss the corresponding Fefferman metri...

It is proved that every locally conformal flat Riemannian manifold all of whose Jacobi operators have constant eigenvalues along every geodesic is with constant principal Ricci curvatures. A local classification (up to an isometry) of locally conformal flat Riemannian manifold with constant Ricci eigenvalues is given in dimensions 4, 5, 6, 7 and 8. It is shown that any n-dimensional (4 ≤ n ≤ 8)...

We study Ricci solitons in Lorentzian α-Sasakian manifolds. It is shown that a symmetric parallel second order covariant tensor in a Lorentzian α-Sasakian manifold is a constant multiple of the metric tensor. Using this it is shown that if LV g + 2S is parallel, V is a given vector field then (g, V ) is Ricci soliton. Further, by virtue of this result Ricci solitons for (2n + 1)-dimensional Lor...

The motive in this article is twofold. First we investigate the geometrical structures of a pseudo-symmetric spacetime (PS)4 with timelike vector under condition conformal flatness. We classify it into two possible types: constant Ricci scalar and closed velocity vector. Then further study type solutions F(R)-gravity theory demonstrate that pressure energy density effective cosmological perfect...