Related papers: Hypersurfaces of Product Spaces with a Canonical D…

Given a vector field $X$ in a Riemannian manifold, a hypersurface is said to have a canonical principal direction relative to $X$ if the projection of $X$ onto the tangent space of the hypersurface gives a principal direction. We give…

Given a compact Riemannian manifold $M$, we consider a warped product $\bar M = I \times_h M$ where $I$ is an open interval in $\Rr$. We suppose that the mean curvature of the fibers do not change sign. Given a positive differentiable…

In this paper we study constant mean curvature surfaces $\Sigma$ in a product space, $\mathbb{M}^2\times \mathbb{R}$, where $\mathbb{M}^2$ is a complete Riemannian manifold. We assume the angle function $\nu = \meta{N}{\partial_t}$ does not…

Given a compact Riemannian manifold $M$, we consider a warped product $\bar M = I \times_h M$ where $I$ is an open interval in $\Bbb R$. For a positive function $\psi$ defined on $\bar M$, we generalized the arguments in \cite{GRW2015} and…

Given a Riemannian manifold $M,$ and an open interval $I\subset\mathbb{R},$ we characterize nontrivial totally umbilical hypersurfaces of the product $M\times I$ -- as well as of warped products $I\times_\omega M$ -- as those which are…

We first show that every isoparametric hypersurface in $\mathbb{S}^{n}\times \mathbb{R}^{m}$ or $\mathbb{H}^{n}\times \mathbb{R}^{m}$ possesses a constant angle function with respect to the canonical product structure. Exploiting this…

We consider compact hypersurfaces in an $(n+1)$-dimensional either Riemannian or Lorentzian space $N^{n+1}$ endowed with a conformal Killing vector field. For such hypersurfaces, we establish an integral formula which, especially in the…

Let M be a compact Riemannian manifold without boundary and let E be a Riemannian vector bundle over M. If $\Sigma$ denotes the sphere subbundle of E, we look for embeddings of $\Sigma$ into E admitting a prescribed mean curvatures of…

In this paper, almost product-like Riemannian manifolds are investigated. Basic properties on tangential hypersurfaces of almost product-like Riemannian manifolds are obtained. Some examples of tangential hypersurfaces are presented. Some…

In this article, we consider compact surfaces $\Sigma$ having constant mean curvature $H$ ($H$-surfaces) whose boundary $\Gamma=\partial\Sigma\subset \mathbb{M}_0= \mathbb{M} \times_f\{0\}$ is transversal to the slice $\mathbb{M}_0$ of the…

We consider hypersurfaces of products $M\times\mathbb R$ with constant $r$-th mean curvature $H_r\ge 0$ (to be called $H_r$-hypersurfaces), where $M$ is an arbitrary Riemannian $n$-manifold. We develop a general method for constructing…

We give a complete classification of isoparametric hypersurfaces in a product space $M^2_{\kappa_1}\times M^2_{\kappa_2}$ of $2$-dimensional space forms for $\kappa_i\in \{-1,0,1\}$ with $\kappa_1\neq \kappa_2$. In fact we prove that any…

The Riemannian product $\mathbb M_1(c_1) \times \mathbb M_2(c_2)$, where $\mathbb M_i(c_i)$ denotes the $2$-dimensional space form of constant sectional curvature $c_i \in \mathbb R$, has two different Spin$^c$ structures carrying each a…

We prove that the angle function associated with the canonical product structure is constant for an isoparametric hypersurface in $\mathbb{S}^{n}\times \mathbb{S}^{m}$, $\mathbb{S}^{n}\times \mathbb{H}^{m}$, or $\mathbb{H}^{n}\times…

In this paper, we study biharmonic hypersurfaces in a product of an Einstein space and a real line. We prove that a biharmonic hypersurface with constant mean curvature in such a product is either minimal or a vertical cylinder generalizing…

We prove that, in Minkowski space, if a spacelike, $(n-1)$-convex hypersurface $M$ with constant $\sigma_{n-1}$ curvature has bounded principal curvatures, then $M$ is convex. Moreover, if $M$ is not strictly convex, after an…

In this paper, we study \lambda-biharmonic hypersurfaces in the product space L^{m}\times\mathbb{R}, where L^{m} is an Einstein space and \mathbb{R} is a real line. We prove that \lambda-biharmonic hypersurfaces with constant mean curvature…

In this note, we prove that for every $0<\sigma<1$, there exists a smooth complete hypersurface $\Sigma$ in $\mathbb{H}^{n+1}$ with prescribed asymptotic boundary $\partial \Sigma=\Gamma$ at infinity, whose principal curvatures…

We consider Einstein hypersurfaces of warped products $I\times_\omega\mathbb Q_\epsilon^n,$ where $I\subset\mathbb R$ is an open interval and $\mathbb Q_\epsilon^n$ is the simply connected space form of dimension $n\ge 2$ and constant…

We generalize a Bernstein-type result due to Albujer and Al\'ias, for maximal surfaces in a curved Lorentzian product 3-manifold of the form $\Sigma_1\times \mathbb{R}$, to higher dimension and codimension. We consider $M$ a complete…