A characterization of a minimal submanifold in
Themis Koufogiorgos (1983)
Annales Polonici Mathematici
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Themis Koufogiorgos (1983)
Annales Polonici Mathematici
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Dario Di Pinto, Antonio Lotta (2023)
Archivum Mathematicum
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We prove a Frankel type theorem for 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 submanifolds of Sasakian space forms.
Ximin Liu (2002)
Archivum Mathematicum
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Let be a Riemannian -manifold. Denote by and the Ricci tensor and the maximum Ricci curvature on , respectively. In this paper we prove that every totally real submanifolds of a quaternion projective space satisfies , where and are the square mean curvature function and metric tensor on , respectively. The equality holds identically if and only if either is totally geodesic submanifold or and is totally umbilical submanifold. Also we show that if a Lagrangian submanifold...
Minoru Kobayashi (1987)
Colloquium Mathematicae
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Ze-Jun Hu, Guo-Xin Wei (2003)
Colloquium Mathematicae
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Let M̅ be a compact Riemannian manifold with sectional curvature satisfying (resp. ), 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.
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 -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 of index , 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...
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 . By using the Sobolev inequalities of P. Li to get estimates for the norms of certain tensors related to the second fundamental form of M, we prove some rigidity theorems. Denote by H and the mean curvature and the norm of the square length of the second fundamental form of M. We show that there is a constant C such that if , then M is a minimal submanifold in the sphere with sectional...
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...
Alan Weinstein (2000)
Journal of the European Mathematical Society
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We define a distance between submanifolds of a riemannian manifold and show that, if a compact submanifold is not moved too much under the isometric action of a compact group , there is a -invariant submanifold -close to . 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...
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, -projectively flat, -projectively flat, --projectively flat, pseudo projectively flat and -pseudo projectively flat anti-invariant submanifolds of trans-Sasakian manifold admitting Zamkovoy connection are discussed. Moreover, Ricci solitons on...
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 with flat normal bundle. We prove that if the second fundamental form A of M satisfies , where α ∈ [2(1 - √(2/n)), 2(1 + √(2/n))], then M is an affine n-dimensional plane. In particular, if n ≤ 8 and , d = 1,3, then M is an affine n-dimensional plane. Moreover, complete strongly stable hypersurfaces with constant mean curvature and finite -norm curvature in ℝ⁷ are considered.
Shyamal K. Hui, Richard S. Lemence, Pradip Mandal (2020)
Commentationes Mathematicae Universitatis Carolinae
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A submanifold of a generalized Sasakian-space-form is said to be -totally real submanifold if and for all . In particular, if , then 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.
Miroslava Petrović-Torgašev, Leopold C. A. Verstraelen (2008)
Archivum Mathematicum
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It was conjectured in [26] that, for all submanifolds of all real space forms , the Wintgen inequality is valid at all points of , whereby is the normalised scalar curvature of the Riemannian manifold and , respectively , are the squared mean curvature and the normalised scalar normal curvature of the submanifold in the ambient space , and this conjecture was shown there to be true whenever codimension . For a given Riemannian manifold , this inequality can be interpreted...