### On Ricci curvature of $C$-totally real submanifolds in Sasakian space forms.

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We prove some pinching theorems with respect to the scalar curvature of 4-dimensional conformally flat (concircularly flat, quasi-conformally flat) totally real minimal submanifolds in QP⁴(c).

Let ${M}^{n}$ be a Riemannian $n$-manifold. Denote by $S\left(p\right)$ and $\overline{Ric}\left(p\right)$ the Ricci tensor and the maximum Ricci curvature on ${M}^{n}$, respectively. In this paper we prove that every totally real submanifolds of a quaternion projective space $Q{P}^{m}\left(c\right)$ satisfies $S\le ((n-1)c+\frac{{n}^{2}}{4}{H}^{2})g$, where ${H}^{2}$ and $g$ are the square mean curvature function and metric tensor on ${M}^{n}$, respectively. The equality holds identically if and only if either ${M}^{n}$ is totally geodesic submanifold or $n=2$ and ${M}^{n}$ is totally umbilical submanifold. Also we show that if a Lagrangian submanifold of...

We consider an almost Kenmotsu manifold ${M}^{2n+1}$ with the characteristic vector field ξ belonging to the (k,μ)’-nullity distribution and h’ ≠ 0 and we prove that ${M}^{2n+1}$ is locally isometric to the Riemannian product of an (n+1)-dimensional manifold of constant sectional curvature -4 and a flat n-dimensional manifold, provided that ${M}^{2n+1}$ is ξ-Riemannian-semisymmetric. Moreover, if ${M}^{2n+1}$ is a ξ-Riemannian-semisymmetric almost Kenmotsu manifold such that ξ belongs to the (k,μ)-nullity distribution, we prove that ${M}^{2n+1}$ is...

A new class of $(n+1)$-dimensional Lorentz spaces of index $1$ is introduced which satisfies some geometric conditions and can be regarded as a generalization of Lorentz space form. Then, the compact space-like hypersurface with constant scalar curvature of this spaces is investigated and a gap theorem for the hypersurface is obtained.

In this paper, we study the stability of space-like hypersurfaces with constant scalar curvature immersed in the de Sitter spaces.

In this paper, by using Cheng-Yau’s self-adjoint operator $\square $, we study the complete hypersurfaces in a sphere with constant scalar curvature.

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