arXiv research

Classifies and describes hypersurfaces in Siklos spacetimes.

Classifies Calabi hypersurfaces with parallel Fubini-Pick form.

Classifies special hypersurfaces in Gödel spacetimes.

Study of convex hypersurfaces with specific curvature properties.

Paper proves non-existence of certain hypersurfaces in complex quadric.

The study finds necessary and sufficient conditions for $C^1$-hypersurfaces to have nowhere $C^1$-regular parallel sets.

This paper focuses on the study of three dimensional real hypersurfaces in non-flat complex space forms whose $^{*}$-Ricci tensor satisfies conditions of parallelism. More precisely, extension of existing results concerning real hypersurfaces with vanishing, semi-parallel and pseudo-parallel $^{*}$-Ricci tensor in case…

Study of hypersurfaces in Sol4_0 geometry, classifying parallel and totally umbilical types.

Paper studies second order symmetric parallel tensors in generalized f.pk-space forms.

Paper classifies special Euclidean hypersurfaces with specific geometric properties.

The study characterizes canal hypersurfaces formed by non-null curves with parallel frame in Minkowski space-time.

First we introduce the notion of parallel structure Jacobi operator for real hypersurfaces in the complex quadric $Q^m = SO_{m+2}/SO_mSO_2$ . Next we give a complete classification of real hypersurfaces in $Q^m = SO_{m+2}/SO_mSO_2$ with parallel structure Jacobi operator.

In this paper, we study locally strongly convex centroaffine hypersurfaces with parallel cubic form with respect to the Levi-Civita connection of the centroaffine metric. As the main result, we obtain a complete classification of such centroaffine hypersurfaces. The result of this paper is a centroaffine version of the…

We introduce the notion of Reeb parallel structure Jacobi operator for real hypersurfaces in the complex hyperbolic quadric ${Q^*}^m=SO^0_{2,m}/SO_2 SO_m$, $m \geq 3$, and give a classification theory for real hypersurfaces in ${{Q^*}^m}$, $m \geq 3$, with Reeb parallel structure Jacobi operator.

There are several kinds of classification problems for real hypersurfaces in complex two-plane Grassmannians $G_2({\mathbb C}^{m+2})$. Among them, Suh classified Hopf hypersurfaces $M$ in $G_2({\mathbb C}^{m+2})$ with Reeb parallel Ricci tensor in Levi-Civita connection. In this paper, we introduce a new notion of gene…

This paper classifies hypersurfaces in n+1 with parallel Fubini-Pick form.

We classify totally geodesic and parallel hypersurfaces of four-dimensional non-reductive homogeneous pseudo-Riemannian manifolds.

We prove that there does not exist any semi-parallel real hypersurface in complex two-plane Grassmannians. With this result, the nonexistence of recurrent real hypersurfaces in complex two-plane Grassmannians can also be proved.

We classify real hypersurfaces in CP^2and CH^2 equipped with pseudo-parallel structure Jacobi operator.

We study three dimensional real hypersurfaces in CP^2 and CH^2 equipped with $xi$-parallel structure Jacobi operator. We prove that they are Hopf hypersurfaces and if additional $α\neq0$, we classify them.

Study proves no lightlike hypersurfaces exist in certain indefinite Sasakian manifolds.

The objective of the present paper is to prove the non-existence of real hypersurface with pseudo-parallel normal Jacobi operator in complex two-plane Grassmannians. As a corollary, we show that there does not exist any real hypersurface with semi-parallel or recurrent normal Jacobi operator in complex two-plane Grassm…

We prove that there does not exist any real hypersurface in complex Grassmannians of rank two with semi-parallel structure Jacobi operator. With this result, the nonexistence of real hypersurface in complex Grassmannians of rank two with recurrent structure Jacobi operator is proved.

We prove the non-existence of real hypersurfaces in CP^2 and CH^2 whose structure Jacobi operator is Lie D-parallel.

Formula for spacelike submanifolds in warped products.

It is shown that a hypersurface of a space form is the initial data for a solution to the mean curvature flow by parallel hypersurfaces if, and only if, it is isoparametric. By solving an ordinary differential equation, explicit solutions are given for all isoparametric hypersurfaces of space forms. In particular, for …

In 1970, Samuel I. Goldberg and Kentaro Yano defined the notion of noninvariant hypersurface of a Sasakian manifold [1]. In this paper we have studied the properties of parallel vector fields with respect to induced connection on the noninvariant hypersurface $M$ of a Sasakian manifold $\tilde M$ with $(φ, g, u, v, λ)-…

In this paper, we introduce new notions of semi-parallel shape operators and structure Jacobi operators in complex two-plane Grassmannians $G_2({\mathbb C}^{m+2})$. By using such a semi-parallel condition, we give a complete classification of Hopf hypersurfaces in $G_2({\mathbb C}^{m+2})$.

The paper examines stable CMC hypersurfaces with boundaries on parallel hyperplanes.

Totally umbilical hypersurfaces in Spin^c manifolds with special spinors have constant mean curvature.

In this paper, we give a complete conformal classification of the regular space-like hypersurfaces in the de Sitter Space $\mathbb{S}^{m+1}_{1}$ with parallel para-Blaschke tensors.

The non-existence of three dimensional real hypersurfaces in non-flat complex space forms with parallel *-Ricci tensor is proved.At the end of the papaer ideas for further research on *-Ricci tensor are provided.

We deal with hypersurfaces in the framework of the $n$-dimensional relative differential geometry. We consider a hypersurface $\varPhi$ of $\mathbb{R}^{n+1}$ with position vector field $\mathbf{x}$, which is relatively normalized by a relative normalization $\mathbf{y}$. Then $\mathbf{y}$ is also a relative normalizati…

The paper studies inverse mean curvature flow on hypersurfaces in space forms.

We consider non-degenerate centro-affine hypersurface immersions in R^n whose cubic form is parallel with respect to the Levi-Civita connection of the affine metric. There exists a bijective correspondence between homothetic families of proper affine hyperspheres with center in the origin and with parallel cubic form, …

In this paper, we introduce and study the locally strongly convex equiaffine isoparametric hypersurfaces and equiaffine isoparametric functions on the affine space $A^{n+1}$. Motivated by the case on the Euclidean space $E^{n+1}$, we first introduce the concept of equiaffine parallel hypersurfaces in $A^{n+1}$, obtaini…

We deal with hypersurfaces in the framework of the relative differential geometry in $\mathbb{R}^4$. We consider a hypersurface $\varPhi$ in $\mathbb{R}^4$ with position vector field $\vect{x}$ which is relatively normalized by a relative normalization $\vect{y}$. Then $\vect{y}$ is also a relative normalization of eve…

In this paper, we use two conformal non-homogeneous coordinate systems, modeled on the de Sitter space ${\mathbb S}^{m+1}_1$, to cover the conformal space ${\mathbb Q}^{m+1}_1$, so that the conformal geometry of regular space-like hypersurfaces in $\mathbb{Q}^{m+1}_1$ is treated as that of hypersurfaces in ${\mathbb S}…

Study on real hypersurfaces in products of complex space forms, proving rigidity and nonexistence results.

In this paper, we first introduce the full express of the Riemannian curvature tensor of a real hypersurface $M$ in complex quadric $Q^{m}$ from the equation of Gauss. Next we derive a formula for the structure Jacobi operator $R_ξ$ and its derivative under the Levi-Civita connection of $M$. We give a complete classifi…

Let $\widetilde{J}$ be the canonical para-complex structure on $\mathbb{R}^{2n+2}\simeq\widetilde{\mathbb{C}}^{n+1}$. We study real affine hypersurfaces $f\colon M\rightarrow \widetilde{\mathbb{C}}^{n+1}$ with a $\widetilde{J}$-tangent transversal vector field. Such vector field induces in a natural way an almost parac…

The study defines and constructs hypersurfaces in a product of two space forms.

We explicitly determine the structure equations of 5-dimensional Levi 2-nondegenerate CR hypersurfaces, using our recently constructed canonical Cartan connection for this class of CR manifolds. We also give an outline of the basic properties of absolute parallelisms and Cartan connections, together with a brief discus…

Simply connected 3-dimensional homogeneous manifolds $E(κ, τ)$, with 4-dimensional isometry group, have a canonical Spin$^c$ structure carrying parallel or Killing spinors. The restriction to any hypersurface of these parallel or Killing spinors allows to characterize isometric immersions of surfaces into $E(κ, τ)$. As…

On a Lorentzian manifold the existence of a parallel null vector field implies certain constraint conditions on the induced Riemannian geometry of a space-like hypersurface. We will derive these constraint conditions and, conversely, show that every real analytic Riemannian manifold satisfying the constraint conditions…

In this paper, we prove that minimal hypersurfaces when $n\geq 3$ and nonzero constant mean curvature hypersurfaces when $n\geq2$ foliated by spheres in parallel horizontal hyperplanes in ${\mathbb{H}}^n \times \mathbb{R}$ must be rotationally symmetric.

Study shows lightlike hypersurfaces in indefinite Sasakian manifolds are not symmetric.

We construct a special class of Lorentz surfaces in the pseudo-Euclidean 4-space with neutral metric which are one-parameter systems of meridians of rotational hypersurfaces with lightlike axis and call them meridian surfaces. We give the complete classification of the meridian surfaces with constant Gauss curvature an…