Weingarten hypersurfaces of the spherical type in Euclidean spaces

Cid D. F. Machado; Carlos M. C. Riveros

Displaying similar documents to “Weingarten hypersurfaces of the spherical type in Euclidean spaces”

On Some Family of Contractible Hypersurfaces in $C^{4}$

S. Kaliman, L. Makar-Limanov (1993)

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New characterizations of linear Weingarten hypersurfaces immersed in the hyperbolic space

Cícero P. Aquino, Henrique F. de Lima (2015)

Archivum Mathematicum

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In this paper, we deal with complete linear Weingarten hypersurfaces immersed in the hyperbolic space $ℍ^{n + 1}$ , that is, complete hypersurfaces of $ℍ^{n + 1}$ whose mean curvature $H$ and normalized scalar curvature $R$ satisfy $R = a H + b$ for some $a$ , $b \in ℝ$ . In this setting, under appropriate restrictions on the mean curvature and on the norm of the traceless part of the second fundamental form, we prove that such a hypersurface must be either totally umbilical or isometric to a hyperbolic cylinder of $ℍ^{n + 1}$ . Furthermore,...

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Thomas Hasanis (1984)

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A short note on $f$ -biharmonic hypersurfaces

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Commentationes Mathematicae Universitatis Carolinae

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In the present paper we give some properties of $f$ -biharmonic hypersurfaces in real space forms. By using the $f$ -biharmonic equation for a hypersurface of a Riemannian manifold, we characterize the $f$ -biharmonicity of constant mean curvature and totally umbilical hypersurfaces in a Riemannian manifold and, in particular, in a real space form. As an example, we consider $f$ -biharmonic vertical cylinders in $S^{2} \times ℝ$ .

A characterization of a certain real hypersurface of type $(A_{2})$ in a complex projective space

Byung Hak Kim, In-Bae Kim, Sadahiro Maeda (2017)

Czechoslovak Mathematical Journal

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In the class of real hypersurfaces $M^{2 n - 1}$ isometrically immersed into a nonflat complex space form ${\tilde{M}}_{n} (c)$ of constant holomorphic sectional curvature $c$ $(\neq 0)$ which is either a complex projective space $ℂ P^{n} (c)$ or a complex hyperbolic space $ℂ H^{n} (c)$ according as $c > 0$ or $c < 0$ , there are two typical examples. One is the class of all real hypersurfaces of type (A) and the other is the class of all ruled real hypersurfaces. Note that the former example are Hopf manifolds and the latter are non-Hopf manifolds....

A half-space type property in the Euclidean sphere

Marco Antonio Lázaro Velásquez (2022)

Archivum Mathematicum

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We study the notion of strong $r$ -stability for the context of closed hypersurfaces $Σ^{n}$ ( $n \geq 3$ ) with constant $(r + 1)$ -th mean curvature $H_{r + 1}$ immersed into the Euclidean sphere $𝕊^{n + 1}$ , where $r \in {1, ..., n - 2}$ . In this setting, under a suitable restriction on the $r$ -th mean curvature $H_{r}$ , we establish that there are no $r$ -strongly stable closed hypersurfaces immersed in a certain region of $𝕊^{n + 1}$ , a region that is determined by a totally umbilical sphere of $𝕊^{n + 1}$ . We also provide a rigidity result for such hypersurfaces.

A genericity theorem for algebraic stacks and essential dimension of hypersurfaces

Zinovy Reichstein, Angelo Vistoli (2013)

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We compute the essential dimension of the functors Forms $_{n, d}$ and Hypersurf $_{n, d}$ of equivalence classes of homogeneous polynomials in $n$ variables and hypersurfaces in $ℙ^{n - 1}$ , respectively, over any base field $k$ of characteristic $0$ . Here two polynomials (or hypersurfaces) over $K$ are considered equivalent if they are related by a linear change of coordinates with coefficients in $K$ . Our proof is based on a new Genericity Theorem for algebraic stacks, which is of independent interest. As another application...

Spacelike intersection curve of three spacelike hypersurfaces in $E_{1}^{4}$

B. Uyar Duldul, M. Caliskan (2013)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this paper, we compute the Frenet vectors and the curvatures of the spacelike intersection curve of three spacelike hypersurfaces given by their parametric equations in four-dimensional Minkowski space $E_{1}^{4}$ .

A certain tensor on real hypersurfaces in a nonflat complex space form

Kazuhiro Okumura (2020)

Czechoslovak Mathematical Journal

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In a nonflat complex space form (namely, a complex projective space or a complex hyperbolic space), real hypersurfaces admit an almost contact metric structure $(φ, ξ, η, g)$ induced from the ambient space. As a matter of course, many geometers have investigated real hypersurfaces in a nonflat complex space form from the viewpoint of almost contact metric geometry. On the other hand, it is known that the tensor field $h$ $(= \frac{1}{2} ℒ_{ξ} φ)$ plays an important role in contact Riemannian geometry. In this...

A characterization of n-dimensional hypersurfaces in $R^{n + 1}$ with commuting curvature operators

Yulian T. Tsankov (2005)

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Let Mⁿ be a hypersurface in $R^{n + 1}$ . We prove that two classical Jacobi curvature operators $J_{x}$ and $J_{y}$ commute on Mⁿ, n > 2, for all orthonormal pairs (x,y) and for all points p ∈ M if and only if Mⁿ is a space of constant sectional curvature. Also we consider all hypersurfaces with n ≥ 4 satisfying the commutation relation $(K_{x, y} \circ K_{z, u}) (u) = (K_{z, u} \circ K_{x, y}) (u)$ , where $K_{x, y} (u) = R (x, y, u)$ , for all orthonormal tangent vectors x,y,z,w and for all points p ∈ M.

Hypersurfaces with singular locus a plane curve and transversal type $A_{1}$

Dirk Siersma (1988)

Banach Center Publications

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Lightlike hypersurfaces of an indefinite Kaehler manifold of a quasi-constant curvature

Dae Ho Jin, Jae Won Lee (2019)

Communications in Mathematics

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We study lightlike hypersurfaces $M$ of an indefinite Kaehler manifold $\bar{M}$ of quasi-constant curvature subject to the condition that the characteristic vector field $ζ$ of $\bar{M}$ is tangent to $M$ . First, we provide a new result for such a lightlike hypersurface. Next, we investigate such a lightlike hypersurface $M$ of $\bar{M}$ such that (1) the screen distribution $S (T M)$ is totally umbilical or (2) $M$ is screen conformal.

Hypersurfaces with free boundary and large constant mean curvature: concentration along submanifolds

Mouhamed Moustapha Fall, Fethi Mahmoudi (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Given a domain $Ω$ of $ℝ^{m + 1}$ and a $k$ -dimensional non-degenerate minimal submanifold $K$ of $\partial Ω$ with $1 \leq k \leq m - 1$ , we prove the existence of a family of embedded constant mean curvature hypersurfaces in $Ω$ which as their mean curvature tends to infinity concentrate along $K$ and intersecting $\partial Ω$ perpendicularly along their boundaries.

$f$ -biminimal maps between Riemannian manifolds

Yan Zhao, Ximin Liu (2019)

Czechoslovak Mathematical Journal

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We give the definition of $f$ -biminimal submanifolds and derive the equation for $f$ -biminimal submanifolds. As an application, we give some examples of $f$ -biminimal manifolds. Finally, we consider $f$ -minimal hypersurfaces in the product space $ℝ^{n} \times 𝕊^{1} (a)$ and derive two rigidity theorems.

Real hypersurfaces with an induced almost contact structure

Michał Szancer, Zuzanna Szancer (2009)

Colloquium Mathematicae

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We study real affine hypersurfaces $f : M \to ℂ^{n + 1}$ with an almost contact structure (φ,ξ,η) induced by any J-tangent transversal vector field. The main purpose of this paper is to show that if (φ,ξ,η) is metric relative to the second fundamental form then it is Sasakian and moreover f(M) is a piece of a hyperquadric in $ℝ^{2 n + 2}$ .

Implicitization of Parametric Hypersurfaces via Points

Ferruccio Orecchia, Isabella Ramella (2018)

Rendiconto dell’Accademia delle Scienze Fisiche e Matematiche

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Given a parametric polynomial representation of an algebraic hypersurface $\mathbf{S}$ in the projective space we give a new algorithm for finding the implicit cartesian equation of $\mathbf{S}$ .The algorithm is based on finding a suitable finite number of points on $\mathbf{S}$ and computing, by linear algebra, the equation of the hypersurface of least degree that passes through the points. In particular the algorithm works for plane curves and surfaces in the ordinary three-dimensional space. Using C++ the algorithm...

$J$ -holomorphic discs and real analytic hypersurfaces

William Alexandre, Emmanuel Mazzilli (2014)

Annales de l’institut Fourier

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We give in $ℝ^{6}$ a real analytic almost complex structure $J$ , a real analytic hypersurface $M$ and a vector $v$ in the Levi null set at $0$ of $M$ , such that there is no germ of $J$ -holomorphic disc $γ$ included in $M$ with $γ (0) = 0$ and $\frac{\partial γ}{\partial x} (0) = v$ , although the Levi form of $M$ has constant rank. Then for any hypersurface $M$ and any complex structure $J$ , we give sufficient conditions under which there exists such a germ of disc.

Pointed $k$ -surfaces

Graham Smith (2006)

Bulletin de la Société Mathématique de France

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Let $S$ be a Riemann surface. Let $ℍ^{3}$ be the $3$ -dimensional hyperbolic space and let $\partial_{\infty} ℍ^{3}$ be its ideal boundary. In our context, a Plateau problem is a locally holomorphic mapping $ϕ : S \to \partial_{\infty} ℍ^{3} = \hat{ℂ}$ . If $i : S \to ℍ^{3}$ is a convex immersion, and if $N$ is its exterior normal vector field, we define the Gauss lifting, $\hat{ı}$ , of $i$ by $\hat{ı} = N$ . Let $\vec{n} : U ℍ^{3} \to \partial_{\infty} ℍ^{3}$ be the Gauss-Minkowski mapping. A solution to the Plateau problem $(S, ϕ)$ is a convex immersion $i$ of constant Gaussian curvature equal to $k \in (0, 1)$ such that the Gauss lifting $(S, \hat{ı})$ is complete and $\vec{n} \circ \hat{ı} = ϕ$ . In this...