Real hypersurfaces in complex hyperbolic two-plane Grassmannians with commuting restricted normal Jacobi operators

Doo Hyun Hwang; Eunmi Pak; Changhwa Woo

Displaying similar documents to “Real hypersurfaces in complex hyperbolic two-plane Grassmannians with commuting restricted normal Jacobi operators”

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,...

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

Yulian T. Tsankov (2005)

Banach Center Publications

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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.

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 genericity theorem for algebraic stacks and essential dimension of hypersurfaces

Zinovy Reichstein, Angelo Vistoli (2013)

Journal of the European Mathematical Society

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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...

A short note on $f$ -biharmonic hypersurfaces

Selcen Y. Perktaş, Bilal E. Acet, Adara M. Blaga (2020)

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 $S^{2} \times ℝ$ .

Generalized Tanaka-Webster and Levi-Civita connections for normal Jacobi operator in complex two-plane Grassmannians

Eunmi Pak, Juan de Dios Pérez, Young Jin Suh (2015)

Czechoslovak Mathematical Journal

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We study classifying problems of real hypersurfaces in a complex two-plane Grassmannian $G_{2} (ℂ^{m + 2})$ . In relation to the generalized Tanaka-Webster connection, we consider that the generalized Tanaka-Webster derivative of the normal Jacobi operator coincides with the covariant derivative. In this case, we prove complete classifications for real hypersurfaces in $G_{2} (ℂ^{m + 2})$ satisfying such conditions.

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 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.

Fundamental solution $E_{n}$ of the operator $\partial^{2} / \partial t^{2} - Δ_{n}$ for n>3

Zofia Szmydt, Bogdan Ziemian (1979)

Annales Polonici Mathematici

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

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

Cours de l'institut Fourier

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Almost everywhere convergence of the inverse Jacobi transform and endpoint results for a disc multiplier

Troels Roussau Johansen (2011)

Studia Mathematica

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The maximal operator S⁎ for the spherical summation operator (or disc multiplier) $S_{R}$ associated with the Jacobi transform through the defining relation $\hat{S_{R} f} (λ) = 1_{| λ | \leq R} f ̂ (t)$ for a function f on ℝ is shown to be bounded from $L^{p} (ℝ ₊, d μ)$ into $L^{p} (ℝ, d μ) + L ² (ℝ, d μ)$ for (4α + 4)/(2α + 3) < p ≤ 2. Moreover S⁎ is bounded from $L^{p ₀, 1} (ℝ ₊, d μ)$ into $L^{p ₀, \infty} (ℝ, d μ) + L ² (ℝ, d μ)$ . In particular ${S_{R} f (t)}_{R > 0}$ converges almost everywhere towards f, for $f \in L^{p} (ℝ ₊, d μ)$ , whenever (4α + 4)/(2α + 3) < p ≤ 2.

Hyperideal polyhedra in hyperbolic 3-space

Xiliang Bao, Francis Bonahon (2002)

Bulletin de la Société Mathématique de France

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A hyperideal polyhedron is a non-compact polyhedron in the hyperbolic $3$ -space $ℍ^{3}$ which, in the projective model for $ℍ^{3} \subset {ℝℙ}^{3}$ , is just the intersection of $ℍ^{3}$ with a projective polyhedron whose vertices are all outside $ℍ^{3}$ and whose edges all meet $ℍ^{3}$ . We classify hyperideal polyhedra, up to isometries of $ℍ^{3}$ , in terms of their combinatorial type and of their dihedral angles.

The existence of Carathéodory solutions of hyperbolic functional differential equations

Adrian Karpowicz (2010)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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We consider the following Darboux problem for the functional differential equation $\partial ² u / \partial x \partial y (x, y) = f (x, y, u_{(x, y)}, \partial u / \partial x (x, y), \partial u / \partial y (x, y))$ a.e. in [0,a]×[0,b], u(x,y) = ψ(x,y) on [-a₀,a]×[-b₀,b] $0, a] \times (0, b],$ where the function $u_{(x, y)} : [- a ₀, 0] \times [- b ₀, 0] \to ℝ^{k}$ is defined by $u_{(x, y)} (s, t) = u (s + x, t + y)$ for (s,t) ∈ [-a₀,0]×[-b₀,0]. We prove a theorem on existence of the Carathéodory solutions of the above problem.

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.

A multiparameter variant of the Salem-Zygmund central limit theorem on lacunary trigonometric series

Mordechay B. Levin (2013)

Colloquium Mathematicae

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We prove the central limit theorem for the multisequence $\sum_{1 \leq n ₁ \leq N ₁} \dots \sum_{1 \leq n_{d} \leq N_{d}} a_{n ₁, . . ., n_{d}} c o s (⟨ 2 π m, A ₁^{n ₁} . . . A_{d}^{n_{d}} x ⟩)$ where $m \in ℤ^{s}$ , $a_{n ₁, . . ., n_{d}}$ are reals, $A ₁, . . ., A_{d}$ are partially hyperbolic commuting s × s matrices, and x is a uniformly distributed random variable in ${[0, 1]}^{s}$ . The main tool is the S-unit theorem.

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.