Let M be a real hypersurface of a complex space form with almost contact metric structure (φ, ξ, η, g). In this paper, we study real hypersurfaces in a complex space form whose structure Jacobi operator Rξ = R(·, ξ)ξ is ξ-parallel. In particular, we prove that the condition ∇ξRξ = 0 characterizes the homogeneous real hypersurfaces of type A in a complex projective space or a complex hyperbolic ...

It is still an open problem whether Riemannian manifolds all of whose local geodesic symmetries are volume–preserving (i.e., D’Atri spaces) or more generally, ball–homogeneous spaces, and C-spaces are locally homogeneous or not. We provide some partial positive answers by proving that five–dimensional locally φ–symmetric spaces can be characterized as Sasakian spaces which are ball–homogeneous ...

where Rαβ(x, t) denotes the Ricci curvature tensor of the metric gαβ(x, t). One of the main problems in differential geometry is to find canonical structure on manifolds. The Ricci flow introduced by Hamilton [8] is an useful tool to approach such problems. For examples, Hamilton [10] and Chow [7] used the convergence of the Ricci flow to characterize the complex structures on compact Riemann s...

In this paper, we are interested in certain global geometric quantities associated to the Schouten tensor and their relationship in conformal geometry. For an oriented compact Riemannian manifold (M,g) of dimension n > 2, there is a sequence of geometric functionals arising naturally in conformal geometry, which were introduced by Viaclovsky in [29] as curvature integrals of Schouten tensor. If...

Cosmological implications of a generalized Rastall theory where the non-conservation equation is promoted to accepts arbitrary functions Ricci scalar considered. We have obtain model parameters power-law ansatz $f\propto R^\eta$, using $H_0$ and $f\sigma_8$ data sets show that could satisfy observational data. The dynamical analysis shows for $\eta\in(0,1)$, dust dominated fixed point lost as r...

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...

Recently, we have used the symmetric bracket of vector fields, and developed the notion of the symmetric derivation. Using this machinery, we have defined the concept of symmetric curvature. This concept is natural and is related to the notions divergence and Laplacian of vector fields. This concept is also related to the derivations on the algebra of symmetric forms which has been discu...

Suppose that (Mn, g) is an asymptotically flat Riemannian spin manifold of positive scalar curvature. The positive mass theorem [1, 2, 3] states that the total mass of the manifold is always positive, and is zero if and only if the manifold is flat. This result suggests that there should be an inequality which bounds the Riemann tensor in terms of the total mass and implies that curvature must ...

We consider spacetimes consisting of a manifold with Lorentzian metric and a weight function or scalar field. These spacetimes admit a Bakry–Émery–Ricci tensor which is a natural generalization of the Ricci tensor. We impose an energy condition on the Bakry–Émery–Ricci tensor and obtain singularity theorems of a cosmological type, both for zero and for positive cosmological constant. That is, w...

recently, we have used the symmetric bracket of vector fields, and developed the notion of the symmetric derivation. using this machinery, we have defined the concept of symmetric curvature. this concept is natural and is related to the notions divergence and laplacian of vector fields. this concept is also related to the derivations on the algebra of symmetric forms which has been discus...