arXiv research

The study characterizes submanifolds in product spaces.

The study proves that pseudo-umbilical spacelike submanifolds are totally geodesic and totally umbilical.

Totally geodesic Lagrangian submanifolds in nearly Kähler S³×S³.

Minimal totally real submanifolds in complex space forms have special umbilical properties.

Study umbilical properties of spacelike 2D submanifolds in semi-Riemannian geometry.

Study on umbilical submanifolds in specific geometric spaces.

Defines conditions for umbilical submanifolds in arbitrary dimensions.

Study classifies submanifolds in probability simplex.

The purpose of this paper is to classify totally umbilical slant submanifolds of a Kenmotsu manifold. We prove that a totally umbilical slant submanifold $M$ of a Kenmotsu manifold $\bar M$ is either invariant or anti-invariant or $dim M=1$ or the mean curvature vector $H$ of $M$ lies in the invariant normal subbundle.…

The notion of Lagrangian $H$-umbilical submanifolds was introduced by B. Y. Chen in 1997, and these submanifolds have appeared in several important problems in the study of Lagrangian submanifolds from the Riemannian geometric point of view. Recently, the author introduced the notion of tangentially biharmonic submanif…

Classifies special submanifolds in geometric spaces.

The paper classifies submanifolds in pseudo-Riemannian space forms.

The paper studies minimal submanifolds with specific curvature properties in Euclidean space.

Study of spacelike submanifolds with umbilical lightlike normals in Lorentzian spacetimes.

Classification of umbilical submanifolds in a specific Riemannian product space.

The authors obtained the classification theorem for Lagrangian H-umbilical submanifolds in quaternion Euclidean spaces using the idea of quaternion extensors.

Study on hemi-slant submanifolds in nearly Kaehler manifolds.

New classification of spherical 2-Dupin submanifolds.

Using the idea of special Legendre curves, the authors obtained the explicit description of flat Lagrangian H-umbilical submanifolds in quaternion Euclidean spaces.

Study on null helices in semi-Riemannian manifolds with special submanifolds.

New inequality shows all special submanifolds in light cone are totally umbilical spheres.

Study characterizes geometric properties of QGCR-null submanifolds in cosymplectic manifolds.

In [2] we have classified the Blaschke quasi-umbilical submanifolds in the conformal space ${\mathbb Q}^n_s$. In this paper we shall classify the Blaschke para-umbilical hypersurfaces in the conformal space ${\mathbb Q}^n_s$. That may be also considered as the extension of the classification of the conformal isotropic …

Study on CR-lightlike submanifolds in golden semi-Riemannian manifolds.

Totally geodesic submanifolds in spheres have restricted curvature properties.

Study of holomorphic submanifolds in hypercomplex manifolds with specific metrics.

Paper studies CR submanifolds in complex projective space with flat normal connection.

Study calculates total mean curvatures of specific submanifolds.

This paper classifies flat submanifolds with a special type of curvature form.

Given the Lagrangian fibration $T^4\to T^2$ and a Lagrangian submanifold, exhibiting an elliptic umbilic and supporting a flat line bundle, we study, in the context of mirror symmetry, the ``quantum'' corrections necessary to solve the monodromy of the holomorphic structure of the mirror bundle on the dual fibration.

The paper classifies submanifolds in a specific Lorentz-Minkowski space.

The paper studies the affine focal set of submanifolds in hypersurfaces, describing conditions for regularity and singularities.

We give a complete classification of umbilical submanifolds of arbitrary dimension and codimension of $\Sf^n\times \R$, extending the classification of umbilical surfaces in $\Sf^2\times \R$ by Rabah-Souam and Toubiana as well as the local description of umbilical hypersurfaces in $\Sf^n\times \R$ by Van der Veken and …

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 $\Q_{k_1}^{n_1}\times \Q_{k_2}^{n_2}$ of two space forms whose curvatures satisfy $k_1+k_2\neq 0$ to …

We give necessary and sufficient conditions for Riemannian maps to be biharmonic. We also define pseudo umbilical Riemannian maps as a generalization of pseudo-umbilical submanifolds and show that such Riemannian maps put some restrictions on the base manifolds.

Study totally umbilic submanifolds using planar pseudo-geodesics.

Study of CR-submanifolds in various Lorentzian manifolds.

We call indexed-biharmonic maps, the solutions of a particular non linear elliptic PDE of order 4. This is a generalization of harmonic maps which verifies that biharmonic maps are biharmonic of index 0. The goal of this article is to study submanifolds of $\mathbb{S}^n$ whose inclusion is non harmonic and indexed-biha…

Several characterizations of umbilic points of submanifolds in arbitrary Riemannian and Lorentzian manifolds are given. As a consequence, we obtain new characterizations of spheres in the Euclidean space and of hyperbolic spaces in the Lorentz-Minkowski space. We also prove the Lorentzian version of a classical result …

The paper estimates eigenvalues of submanifolds in symmetric spaces.

We classify biharmonic submanifolds with certain geometric properties in Euclidean spheres. For codimension 1, we determine the biharmonic hypersurfaces with at most two distinct principal curvatures and the conformally flat biharmonic hypersurfaces. We obtain some rigidity results for pseudo-umbilical biharmonic subma…

A structure on an almost contact metric manifold is defined as a generalization of well-known cases: Sasakian, quasi-Sasakian, Kenmotsu and cosymplectic. Then we consider a semi-invariant $ξ^{\bot}$-submanifold of a manifold endowed with such a structure and two topics are studied: the integrability of distributions de…

Study of $f$-biharmonic hypersurfaces in conformally flat spaces.

Rigidity for 4D Willmore submanifolds with boundary.

Degenerate submanifolds of pseudo-Riemannian manifolds are quite difficult to study because there is no prefered connection when the submanifold is not totally geodesic. For the particular case of degenerate totally umbilical hypersurfaces, we show that there are "Weyl" connections adapted to the induced structure on t…

In this article we study isometric immersions of nearly Kähler manifolds into a space form (specially Euclidean space) and show that every nearly Kähler submanifold of a space form has a totally umbilic foliation whose leafs are 6-dimensional nearly Kähler manifolds. Moreover using this foliation we show that there is …

In this paper, we derived biharmonic equations for pseudo-Riemannian submanifolds of pseudo-Riemannian manifolds which includes the biharmonic equations for submanifolds of Riemannian manifolds as a special case. As applications, we proved that a pseudo-umbilical biharmonic pseudo-Riemannian submanifold of a pseudo-Rie…

Introduces new lightlike submanifolds in indefinite Kaehler manifolds.