Related papers: Jacobi operators along the structure flow on real …

A Lie hypersurface in the complex hyperbolic space is a homogeneous real hypersurface without focal submanifolds. The set of all Lie hypersurfaces in the complex hyperbolic space is bijective to a closed interval, which gives a deformation…

In this paper, we investigate the spectral properties of the Jacobi operator for immersed surfaces with nonpositive Euler characteristic, extending previous results in the field. We first prove a sharp upper bound for the second eigenvalue…

In the present paper, we study an extended theory of statistical manifolds in application to affine differential geometry. Any smooth hypersurface $M \subset \mathbb{R}^{n+1}$ with a transverse vector field $\xi$ naturally admits a…

In this work we characterize certain immersed closed hypersurfaces of some ambient manifolds via the second eigenvalue of the Jacobi operator. First, we characterize the Clifford torus as the surface which maximizes the second eigenvalue of…

We establish an inequality among the Ricci curvature, the squared mean curvature, and the normal curvature for real hypersurfaces in complex space forms. We classify real hypersurfaces in two-dimensional non-flat complex space forms which…

In this article, we study the quasi-homogeneity of a superpotential in a complete free algebra over an algebraic closed field of characteristic zero. We prove that a superpotential with finite dimensional Jacobi algebra is right equivalent…

In this paper we study $4$-dimensional affine hypersurfaces with a Lorentzian second fundamental form additionally equipped with an almost symplectic structure $\omega$. We prove that the rank of the shape operator is at most one if…

The regular type of a real hyper-surface M in an (almost) complex manifold at some point p is the maximal contact order at p of M with germs of non singular (pseudo) holomorphic disks. The main purpose of this paper is to give two intrinsic…

Let M be a CR submanifold of maximal CR dimension of a complex space form M. The shape operator A of the distinguished vector field {\xi} is recurrent if there exists a 1-form v such that \nabla A = A \otimes v. We show that M is an…

Let $x:M\to\mathbb{S}^{n+1}(1)$ be an n-dimensional compact hypersurface with constant scalar curvature $n(n-1)r,~r\geq 1$, in a unit sphere $\mathbb{S}^{n+1}(1),~n\geq 5$. We know that such hypersurfaces can be characterized as critical…

Let $M$ be a real hypersurface of a complex space form $M^n(c)$, $c\neq 0$. Suppose that the structure vector field $\xi$ of $M$ is an eigen vector field of the Ricci tensor $S$, $S\xi=\beta\xi$, $\beta$ being a function. We study on $M$, a…

Let $M$ be a real hypersurface of a complex space form $M^n(c)$, $c\neq0$, $n\geq 3$. We show that the Ricci tensor $S$ of $M$ satisfies $S(X,Y)=ag(X,Y)$ for any vector fields $X$ and $Y$ on the holomorphic distribution, $a$ being a…

In this paper we first introduce the full expression of the curvature tensor of a real hypersurface $M$ in complex hyperbolic two-plane Grassmannians $SU_{2,m}/S(U_2{\cdot}U_m)$, $m{\ge}2$ from the equation of Gauss. Next we derive a new…

We classify real hypersurfaces in complex space forms with constant principal curvatures and whose Hopf vector field has two nontrivial projections onto the principal curvature spaces. In complex projective spaces such real hypersurfaces do…

We introduce the notion of commuting Ricci tensor for real hypersurfaces in the complex quadric $Q^m = SO_{m+2}/SO_mSO_2$ . It is shown that the commuting Ricci tensor gives that the unit normal vector field $N$ becomes $\frak A$-principal…

The study of real hypersurfaces in pseudo-Riemannian complex space forms and para-complex space forms, which are the pseudo-Riemannian generalizations of the complex space forms, is addressed. It is proved that there are no umbilic…

Let $M$ be a pseudoconvex, oriented, bounded and closed CR submanifold of $\mathbb{C}^{n}$ of hypersurface type. Our main result says that when a certain $1$-form on $M$ is exact on the null space of the Levi form, then the complex Green…

In this article, we introduce the notion of star-Ricci tensors in the real hypersurfaces of complex quadric $Q^m$. It is proved that there exist no Hopf hypersurfaces in $Q^m,m\geq3$, with commuting star-Ricci tensor or parallel star-Ricci…

In a preceding paper we introduced a notion of compatibility between a Jacobi structure and a Riemannian structure on a smooth manifold. We proved that in the case of fundamental examples of Jacobi structures : Poisson structures, contact…

In this article, we get classification theorems for a Ricci soliton on the pseudo-Riemannian hypersurface of the Minkowski space $\mathbb{E}^4_1$ taking the potential vector field as the tangent component of the position vector of the…