Revisiting linear Weingarten spacelike submanifolds immersed in a locally symmetric semi-Riemannian space

Weiller F. C. Barboza; H. F. de Lima; M. A. Velásquez

Displaying similar documents to “Revisiting linear Weingarten spacelike submanifolds immersed in a locally symmetric semi-Riemannian space”

Symmetric twofold CR submanifolds in a Euclidean space $R^{4 m}$

Minoru Kobayashi (1987)

Colloquium Mathematicae

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Almost invariant submanifolds for compact group actions

Alan Weinstein (2000)

Journal of the European Mathematical Society

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We define a $C^{1}$ distance between submanifolds of a riemannian manifold $M$ and show that, if a compact submanifold $N$ is not moved too much under the isometric action of a compact group $G$ , there is a $G$ -invariant submanifold $C^{1}$ -close to $N$ . The proof involves a procedure of averaging nearby submanifolds of riemannian manifolds in a symmetric way. The procedure combines averaging techniques of Cartan, Grove/Karcher, and de la Harpe/Karoubi with Whitney’s idea of realizing submanifolds as zeros...

On the nonexistence of stable minimal submanifolds and the Lawson-Simons conjecture

Ze-Jun Hu, Guo-Xin Wei (2003)

Colloquium Mathematicae

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Let M̅ be a compact Riemannian manifold with sectional curvature $K_{M ̅}$ satisfying $1 / 5 < K_{M ̅} \leq 1$ (resp. $2 \leq K_{M ̅} < 10$ ), which can be isometrically immersed as a hypersurface in the Euclidean space (resp. the unit Euclidean sphere). Then there exist no stable compact minimal submanifolds in M̅. This extends Shen and Xu’s result for 1/4-pinched Riemannian manifolds and also suggests a modified version of the well-known Lawson-Simons conjecture.

Global pinching theorems for minimal submanifolds in spheres

Kairen Cai (2003)

Colloquium Mathematicae

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Let M be a compact submanifold with parallel mean curvature vector embedded in the unit sphere $S^{n + p} (1)$ . By using the Sobolev inequalities of P. Li to get $L_{p}$ estimates for the norms of certain tensors related to the second fundamental form of M, we prove some rigidity theorems. Denote by H and ${| | σ | |}_{p}$ the mean curvature and the $L_{p}$ norm of the square length of the second fundamental form of M. We show that there is a constant C such that if ${| | σ | |}_{n / 2} < C$ , then M is a minimal submanifold in the sphere $S^{n + p - 1} (1 + H ²)$ with sectional...

On some properties of a Riemannian space $V_{4}$ with constant scalar curvature

W. Slósarska, Z. Żekanowski (1972)

Colloquium Mathematicae

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On Deszcz symmetries of Wintgen ideal submanifolds

Miroslava Petrović-Torgašev, Leopold C. A. Verstraelen (2008)

Archivum Mathematicum

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It was conjectured in [26] that, for all submanifolds $M^{n}$ of all real space forms ${\tilde{M}}^{n + m} (c)$ , the Wintgen inequality $ρ \leq H^{2} - ρ^{⊥} + c$ is valid at all points of $M$ , whereby $ρ$ is the normalised scalar curvature of the Riemannian manifold $M$ and $H^{2}$ , respectively $ρ^{⊥}$ , are the squared mean curvature and the normalised scalar normal curvature of the submanifold $M$ in the ambient space $\tilde{M}$ , and this conjecture was shown there to be true whenever codimension $m = 2$ . For a given Riemannian manifold $M$ , this inequality can be interpreted...

A Weitzenbôck formula for the second fundamental form of a Riemannian foliation

Paolo Piccinni (1984)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Si considera la seconda forma fondamentale $\alpha$ di foliazioni su varietà riemanniane e si ottiene una formula per il laplaciano $\nabla^{2}\alpha$ - Se ne deducono alcune implicazioni per foliazioni su varietà a curvatura costante.

Wintgen inequalities on Legendrian submanifolds of generalized Sasakian-space-forms

Shyamal K. Hui, Richard S. Lemence, Pradip Mandal (2020)

Commentationes Mathematicae Universitatis Carolinae

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A submanifold $M^{m}$ of a generalized Sasakian-space-form ${\bar{M}}^{2 n + 1} (f_{1}, f_{2}, f_{3})$ is said to be $C$ -totally real submanifold if $ξ \in Γ (T^{⊥} M)$ and $φ X \in Γ (T^{⊥} M)$ for all $X \in Γ (T M)$ . In particular, if $m = n$ , then $M^{n}$ is called Legendrian submanifold. Here, we derive Wintgen inequalities on Legendrian submanifolds of generalized Sasakian-space-forms with respect to different connections; namely, quarter symmetric metric connection, Schouten-van Kampen connection and Tanaka-Webster connection.

Complete noncompact submanifolds with flat normal bundle

Hai-Ping Fu (2016)

Annales Polonici Mathematici

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Let Mⁿ (n ≥ 3) be an n-dimensional complete super stable minimal submanifold in $ℝ^{n + p}$ with flat normal bundle. We prove that if the second fundamental form A of M satisfies $\int_{M} i {| A |}^{α} < \infty$ , where α ∈ [2(1 - √(2/n)), 2(1 + √(2/n))], then M is an affine n-dimensional plane. In particular, if n ≤ 8 and $\int_{M} {| A |}^{d} < \infty$ , d = 1,3, then M is an affine n-dimensional plane. Moreover, complete strongly stable hypersurfaces with constant mean curvature and finite $L^{α}$ -norm curvature in ℝ⁷ are considered.

$f$ -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.

Riemannian semisymmetric almost Kenmotsu manifolds and nullity distributions

Yaning Wang, Ximin Liu (2014)

Annales Polonici Mathematici

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We consider an almost Kenmotsu manifold $M^{2 n + 1}$ with the characteristic vector field ξ belonging to the (k,μ)’-nullity distribution and h’ ≠ 0 and we prove that $M^{2 n + 1}$ is locally isometric to the Riemannian product of an (n+1)-dimensional manifold of constant sectional curvature -4 and a flat n-dimensional manifold, provided that $M^{2 n + 1}$ is ξ-Riemannian-semisymmetric. Moreover, if $M^{2 n + 1}$ is a ξ-Riemannian-semisymmetric almost Kenmotsu manifold such that ξ belongs to the (k,μ)-nullity distribution, we prove...

The gap theorems for some extremal submanifolds in a unit sphere

Xi Guo and Lan Wu (2015)

Communications in Mathematics

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Let $M$ be an $n$ -dimensional submanifold in the unit sphere $S^{n + p}$ , we call $M$ a $k$ -extremal submanifold if it is a critical point of the functional $\int_{M} ρ^{2 k} d v$ . In this paper, we can study gap phenomenon for these submanifolds.

Structure of second-order symmetric Lorentzian manifolds

Oihane F. Blanco, Miguel Sánchez, José M. Senovilla (2013)

Journal of the European Mathematical Society

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$𝑆𝑒𝑐𝑜𝑛𝑑 - 𝑜𝑟𝑑𝑒𝑟𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐𝐿𝑜𝑟𝑒𝑛𝑡𝑧𝑖𝑎𝑛𝑠𝑝𝑎𝑐𝑒𝑠$ , that is to say, Lorentzian manifolds with vanishing second derivative $\nabla \nabla R \equiv 0$ of the curvature tensor $R$ , are characterized by several geometric properties, and explicitly presented. Locally, they are a product $M = M_{1} \times M_{2}$ where each factor is uniquely determined as follows: $M_{2}$ is a Riemannian symmetric space and $M_{1}$ is either a constant-curvature Lorentzian space or a definite type of plane wave generalizing the Cahen–Wallach family. In the proper case (i.e., $\nabla R \neq 0$ at some point), the curvature...

Curvature tensors and Ricci solitons with respect to Zamkovoy connection in anti-invariant submanifolds of trans-Sasakian manifold

Payel Karmakar (2022)

Mathematica Bohemica

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The present paper deals with the study of some properties of anti-invariant submanifolds of trans-Sasakian manifold with respect to a new non-metric affine connection called Zamkovoy connection. The nature of Ricci flat, concircularly flat, $ξ$ -projectively flat, $M$ -projectively flat, $ξ$ - $M$ -projectively flat, pseudo projectively flat and $ξ$ -pseudo projectively flat anti-invariant submanifolds of trans-Sasakian manifold admitting Zamkovoy connection are discussed. Moreover, Ricci solitons on...