Displaying similar documents to “On minimal generic submanifolds immersed in S 2 m + 1

A Frankel type theorem for CR submanifolds of Sasakian manifolds

Dario Di Pinto, Antonio Lotta (2023)

Archivum Mathematicum

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We prove a Frankel type theorem for C R submanifolds of Sasakian manifolds, under suitable hypotheses on the index of the scalar Levi forms determined by normal directions. From this theorem we derive some topological information about C R submanifolds of Sasakian space forms.

On Ricci curvature of totally real submanifolds in a quaternion projective space

Ximin Liu (2002)

Archivum Mathematicum

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Let M n be a Riemannian n -manifold. Denote by S ( p ) and Ric ¯ ( p ) the Ricci tensor and the maximum Ricci curvature on M n , respectively. In this paper we prove that every totally real submanifolds of a quaternion projective space Q P m ( c ) satisfies S ( ( n - 1 ) c + n 2 4 H 2 ) g , where H 2 and g are the square mean curvature function and metric tensor on M n , respectively. The equality holds identically if and only if either M n is totally geodesic submanifold or n = 2 and M n is totally umbilical submanifold. Also we show that if a Lagrangian submanifold...

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 ̅ 1 (resp. 2 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.

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

Weiller F. C. Barboza, H. F. de Lima, M. A. Velásquez (2023)

Commentationes Mathematicae Universitatis Carolinae

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In this paper, we deal with n -dimensional complete linear Weingarten spacelike submanifolds immersed with parallel normalized mean curvature vector field and flat normal bundle in a locally symmetric semi-Riemannian space L p n + p of index p > 1 , which obeys some curvature constraints (such an ambient space can be regarded as an extension of a semi-Riemannian space form). Under appropriate hypothesis, we are able to prove that such a spacelike submanifold is either totally umbilical or isometric...

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...

Totally contact umbilical screen-slant and screen-transversal lightlike submanifolds of indefinite Kenmotsu manifold

Payel Karmakar (2024)

Mathematica Bohemica

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We study totally contact umbilical screen-slant lightlike submanifolds and totally contact umbilical screen-transversal lightlike submanifolds of an indefinite Kenmotsu manifold. We prove a characterization theorem of totally contact umbilical screen-slant lightlike submanifolds of an indefinite Kenmotsu manifold. We further prove some results on a totally contact umbilical radical screen-transversal lightlike submanifold of an indefinite Kenmotsu manifold, such as the necessary and...

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...

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...

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 n + p with flat normal bundle. We prove that if the second fundamental form A of M satisfies M i | A | α < , where α ∈ [2(1 - √(2/n)), 2(1 + √(2/n))], then M is an affine n-dimensional plane. In particular, if n ≤ 8 and M | A | d < , 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.

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 M ¯ 2 n + 1 ( f 1 , f 2 , f 3 ) is said to be C -totally real submanifold if ξ Γ ( T M ) and φ X Γ ( T M ) for all X 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.

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 M ˜ n + m ( c ) , the Wintgen inequality ρ 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 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...