Related papers: Blaschke Isoparametric Hypersurfaces in the Confor…

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…

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…

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.

Let $M^n$ be an $n$-dimensional umbilic-free hypersurface in the $(n+1)$-dimensional Lorentzian space form $M^{n+1}_1(c)$. Three basic invariants of $M^n$ under the conformal transformation group of $M^{n+1}_1(c)$ are a $1$-form $C$, called…

The conformal geometry of surfaces in the conformal space $\mathbf Q^n_1$ is studied. We classify the space-like surfaces in $\mathbf Q^n_1$ with vanishing conformal form up to conformal equivalence.

We solve Blaschke's problem for hypersurfaces of dimension $n\geq 3$. Namely, we determine all pairs of Euclidean hypersurfaces $f,\tilde{f}\colon M^n\to\R^{n+1}$ that induce conformal metrics on $M^n$ and envelope a common sphere…

In this paper, we introduce isoparametric functions and isoparametric hypersurfaces in Finsler manifolds and give the necessary and sufficient conditions for a transnormal function to be isoparametric. We then prove that hyperplanes,…

In this paper we introduce the fourth fundamental form for the hypersurfaces in $H^{n+1}$ and the space-like hypersurfaces in $S_{1}^{n+1}$ and discuss the conformality of the normal Gauss maps of the hypersurfaces in $H^{n+1}$ and…

The volume distance from a point p to a convex hypersurface M of the (N+1)-dimensional space is defined as the minimum (N+1)-volume of a region bounded by M and a hyperplane H through the point. This function is differentiable in a…

Hypersurfaces embedded in conformal manifolds appear frequently as boundary data in boundary-value problems in cosmology and string theory. Viewed as the non-null conformal infinity of a spacetime, we consider hypersurfaces embedded in a…

We classify isoparametric hypersurfaces in complex hyperbolic spaces.

We address the problem of determining the hypersurfaces $f\colon M^{n} \to \mathbb{Q}_s^{n+1}(c)$ with dimension $n\geq 3$ of a pseudo-Riemannian space form of dimension $n+1$, constant curvature $c$ and index $s\in \{0, 1\}$ for which…

Biharmonic hypersurfaces in a generic conformally flat space are studied in this paper. The equation of such hypersurfaces is derived and is used to determine the conformally flat metric $f^{-2}\delta_{ij}$ on the Euclidean space…

We study translation minimal hypersurfaces and separable minimal hypersurfaces in the ($n+1$)-space with $2m$-norm.

We classify complete biharmonic surfaces with parallel mean curvature vector field and non-negative Gaussian curvature in complex space forms.

It is constructed a formal normal form, using an iterative normalization procedure, for a large class of Real-Smooth Hypersurfaces in Complex Spaces.

In this paper, we first derive biharmonic equation for conformal hypersurfaces in a generic Riemannian manifold generalizing that for biharmonic hypersurfaces in \cite{Ou1} and that for biharmonic conformal surfaces in \cite{Ou3, Ou2, Ou4}.…

Let $\mathbb Q_{\epsilon_i}^{n_i}$ denote the simply connected space form of dimension $n_i\ge 2$ and constant sectional curvature $\epsilon_i$. We prove that any connected isoparametric hypersurface of $\mathbb…

All hypersurface homogeneous locally rotationally symmetric spacetimes which admit conformal symmetries are determined and the symmetry vectors are given explicitly. It is shown that these spacetimes must be considered in two sets. One set…

We establish that equally-spaced smectic configurations enjoy an infinite-dimensional conformal symmetry and show that there is a natural map between them and null hypersurfaces in maximally-symmetric spacetimes. By choosing the appropriate…