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

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

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

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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...

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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.

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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...

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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.

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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.

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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.

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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...

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Structure of second-order symmetric Lorentzian manifolds

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

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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...