相关论文: On submanifolds with recurrent second fundamental …

The properties of Kaehler submanifolds with recurrent the second fundamental form in spaces of constant holomorphic sectional curvature are being studied in this article.

In this paper, we first study isometric immersions $f: M^n\rightarrow M^{n+k}(c), n\geq 3,$ into space forms with flat normal bundle and constant scalar curvature $R.$ Under a suitable multiplicity condition on the second fundamental form…

The following results are proved: Theorem 1. A totally real semiparallel submanifold of constant curvature with parallel f-structure in the normal bundle of a K\"ahler manifold N is flat or a totally geodesic submanifold of N. Theorem 2. A…

In this paper we study biconservative hypersurfaces $M$ in space forms $\overline M^{n+1}(c)$ with four distinct principal curvatures whose second fundamental form has constant norm. We prove that every such hypersurface has constant mean…

Let $M$ be a real $l$-dimensional minimal submanifold with flat normal connection in a kaehler product manifold $\overline{M}^m\times \overline{M}^n$ where $\overline{M}^m$ and $\overline{M}^n$ are complex $m$-dimensional and complex…

We give a complete classification of submanifolds with parallel second fundamental form of a product of two space forms. We also reduce the classification of umbilical submanifolds with dimension $m\geq 3$ of a product $\Q_{k_1}^{n_1}\times…

We classify Lagrangian submanifolds of complex space forms, whose second fundamental form can be written in a certain way, depending on a real parameter. For some special values of this parameter, the resulting submanifolds are ideal in the…

We prove a converse to well-known results by E. Cartan and J. D. Moore. Let $f\colon M^n_c\to\Q^{n+p}_{\tilde c}$ be an isometric immersion of a Riemannian manifold with constant sectional curvature $c$ into a space form of curvature…

Riemannian manifolds of quasi-constant sectional curvatures (QC-manifolds) are divided into two basic classes: with positive or negative horizontal sectional curvatures. We prove that the Riemannian QC-manifolds with positive horizontal…

We show that a complete $m$-dimensional immersed submanifold $M$ of $\mathbb{R}^{n}$ with $a(M)<1$ is properly immersed and have finite topology, where $a(M)\in [0,\infty]$ is an scaling invariant number that gives the rate that the norm of…

An expression for the first variation of the area functional of the second fundamental form is given for a hypersurface in a semi-Riemannian space. The concept of the "mean curvature of the second fundamental form" is then introduced. Some…

We define hypersurfaces $f\colon M^n\to \mathbb{Q}_{c_1}^{k} \times \mathbb{Q}_{c_2}^{n-k+1}$ in class $\mathcal{A}$ of a product of two space forms as those that have flat normal bundle when regarded as submanifolds of the underlying flat…

Two special Finsler spaces have been introduced and investigated, namely $R^h$-recurrent Finsler space and consircularly recurrent Finsler space. The defining properties of these spaces are formulated in terms of the first curvature tensor…

We continue the study of the geometry and topology of compact submanifolds of arbitrary codimension in space forms that satisfy a pinching condition involving the length of the second fundamental form and the mean curvature. Our primary…

In this article we derive a complete classification of all submanifolds in space forms with codimension two for which the Gauss map is homothetic.

A submanifold of a pseudo-Riemannian manifold is said to have parallel mean curvature vector if the mean curvature vector field H is parallel as a section of the normal bundle. Submanifolds with parallel mean curvature vector are important…

We prove that closed manifolds admitting a generic metric whose sectional curvature is locally quasi-constant are graphs of space forms. In the more general setting of QC spaces where sets of isotropic points are arbitrary, under suitable…

In this paper, we have studied biharmonic hypersurfaces in space form $\bar{M}^{n+1}(c)$ with constant sectional curvature $c$. We have obtained that biharmonic hypersurfaces $M^{n}$ with at most three distinct principal curvatures in…

In this paper, we prove that evry 3-dimensional manifold M is a ?- recurrent N(k)-contact metric manifold if and only if it is flat. Then we classify the ?-recurrent contact metric manifolds of constant curvature. This implies that there…

We prove a Simons type formula for submanifolds with parallel mean curvature vector field in product spaces of type $M^n(c)\times\mathbb{R}$, where $M^n(c)$ is a space form with constant sectional curvature $c$, and then we use it to…