Prescribing the positive mixed scalar curvature of totally geodesic foliations

2013 ◽  
pp. 163-203 ◽  
Author(s):  
Vladimir Rovenski ◽  
Leonid Zelenko
Keyword(s):  
Scalar Curvature ◽  
Totally Geodesic

Discs area-minimizing in mean convex Riemannian n-manifolds

2021 ◽  
pp. 1-22
Author(s):  
Ezequiel Barbosa ◽  
Franciele Conrado
Keyword(s):  
Scalar Curvature ◽  
Mean Curvature ◽  
Higher Dimensions ◽  
Dimensional Torus ◽  
Rigidity Result ◽  
Equality Case ◽  
Totally Geodesic ◽  
Zero Degree ◽  
Area Minimizing ◽  
Mean Convex

In this work, we consider oriented compact manifolds which possess convex mean curvature boundary, positive scalar curvature and admit a map to $\mathbb {D}^{2}\times T^{n}$ with non-zero degree, where $\mathbb {D}^{2}$ is a disc and $T^{n}$ is an $n$ -dimensional torus. We prove the validity of an inequality involving a mean of the area and the length of the boundary of immersed discs whose boundaries are homotopically non-trivial curves. We also prove a rigidity result for the equality case when the boundary is strongly totally geodesic. This can be viewed as a partial generalization of a result due to Lucas Ambrózio in (2015, J. Geom. Anal., 25, 1001–1017) to higher dimensions.

THE SCALAR CURVATURE OF MINIMAL HYPERSURFACES IN A UNIT SPHERE

2007 ◽  
Vol 09 (02) ◽  
pp. 183-200 ◽  
Author(s):  
YOUNG JIN SUH ◽  
HAE YOUNG YANG
Keyword(s):  
Scalar Curvature ◽  
Unit Sphere ◽  
Constant Scalar Curvature ◽  
Minimal Hypersurface ◽  
Second Fundamental Form ◽  
Geodesic Sphere ◽  
Clifford Torus ◽  
Totally Geodesic ◽  
Minimal Hypersurfaces ◽  
The Second Fundamental Form

In this paper, we study n-dimensional compact minimal hypersurfaces in a unit sphere Sn+1(1) and give an answer for S. S. Chern's conjecture. We have shown that [Formula: see text] if S > n, and prove that an n-dimensional compact minimal hypersurface with constant scalar curvature in Sn+1(1) is a totally geodesic sphere or a Clifford torus if [Formula: see text], where S denotes the squared norm of the second fundamental form of this hypersurface.

scholarly journals Normal curvature of minimal submanifolds in a sphere

1997 ◽  
Vol 39 (1) ◽  
pp. 29-33
Author(s):  
Sharief Deshmukh
Keyword(s):  
Scalar Curvature ◽  
Unit Sphere ◽  
Minimal Submanifolds ◽  
Normal Curvature ◽  
Minimal Submanifold ◽  
Rigidity Result ◽  
Totally Geodesic ◽  
Veronese Surface ◽  
Arbitrary Codimension ◽  
Image Position

Simons [5] has proved a pinching theorem for compact minimal submanifolds in a unit sphere, which led to an intrinsic rigidity result. Sakaki [4] improved this result of Simons for arbitrary codimension and has proved that if the scalar curvature S of the minimal submanifold Mn of Sn+P satisfiesthen either Mn is totally geodesic or S= 2/3 in which case n = 2 and M2 is the Veronese surface in a totally geodesic 4-sphere. This result of Sakaki was further improved by Shen [6] but only for dimension n=3, where it is shown that if S>4, then M3 is totally geodesic (cf. Theorem 3, p. 791).

Submanifolds with constant scalar curvature

2002 ◽  
Vol 132 (5) ◽  
pp. 1163-1183 ◽  
Author(s):  
Qing-Ming Cheng
Keyword(s):  
Euclidean Space ◽  
Scalar Curvature ◽  
Unit Sphere ◽  
Constant Scalar Curvature ◽  
Second Fundamental Form ◽  
Totally Geodesic ◽  
The Second Fundamental Form ◽  
Complete Submanifolds ◽  
Image Position ◽  
Generalized Cylinder

In this paper, we study n-dimensional complete submanifolds with constant scalar curvature in the Euclidean space En+p and n-dimensional compact submanifolds with constant scalar curvature in the unit sphere Sn+p(1). We prove that the totally umbilical sphere Sn(r), totally geodesic Euclidean space En and generalized cylinder Sn−1(c) × E1 are the only n-dimensional (n > 2) complete submanifolds Mn with constant scalar curvature n(n − 1)r in the Euclidean space En+p, which satisfy the following condition: where S denotes the squared norm of the second fundamental form of Mn. For compact submanifolds with constant scalar curvature in the unit sphere Sn+p(1), we also obtain a corresponding result (see theorem 1.3).

scholarly journals Totally real submanifolds in a complex projective space

1999 ◽  
Vol 22 (1) ◽  
pp. 205-208 ◽  
Author(s):  
Liu Ximin
Keyword(s):  
Projective Space ◽  
Scalar Curvature ◽  
Ricci Curvature ◽  
Complex Projective Space ◽  
Minimal Submanifold ◽  
Real Submanifolds ◽  
Totally Geodesic ◽  
Totally Real ◽  
Complex Projective

In this paper, we establish the following result: LetMbe ann-dimensional complete totally real minimal submanifold immersed inCPnwith Ricci curvature bounded from below. Then eitherMis totally geodesic orinf  r≤(3n+1)(n−2)/3, whereris the scalar curvature ofM.

Submanifolds with constant scalar curvature

2002 ◽  
Vol 132 (5) ◽  
pp. 1163-1183 ◽  
Author(s):  
Qing-Ming Cheng
Keyword(s):  
Euclidean Space ◽  
Scalar Curvature ◽  
Unit Sphere ◽  
Constant Scalar Curvature ◽  
Second Fundamental Form ◽  
Totally Geodesic ◽  
The Second Fundamental Form ◽  
Complete Submanifolds ◽  
Image Position ◽  
Generalized Cylinder

In this paper, we study n-dimensional complete submanifolds with constant scalar curvature in the Euclidean space En+p and n-dimensional compact submanifolds with constant scalar curvature in the unit sphere Sn+p(1). We prove that the totally umbilical sphere Sn(r), totally geodesic Euclidean space En and generalized cylinder Sn−1(c) × E1 are the only n-dimensional (n > 2) complete submanifolds Mn with constant scalar curvature n(n − 1)r in the Euclidean space En+p, which satisfy the following condition: where S denotes the squared norm of the second fundamental form of Mn. For compact submanifolds with constant scalar curvature in the unit sphere Sn+p(1), we also obtain a corresponding result (see theorem 1.3).

scholarly journals A characterisation of compact minimal hypersurfaces in a unit sphere

1993 ◽  
Vol 47 (2) ◽  
pp. 213-216
Author(s):  
Qing-Ming Cheng
Keyword(s):  
Scalar Curvature ◽  
Ricci Curvature ◽  
Unit Sphere ◽  
Constant Scalar Curvature ◽  
Geodesic Sphere ◽  
Clifford Torus ◽  
Totally Geodesic ◽  
Minimal Hypersurfaces

In this note, we show that the totally geodesic sphere, Clifford torus and Cartan hypersurface are the only compact minimal hypersurfaces in S4(1) with constant scalar curvature if the Ricci curvature is not less than −1.

scholarly journals Complete maximal spacelike surfaces in an anti-de Sitter space {\bf H}^{4}_{2}(c)

2000 ◽  
Vol 42 (1) ◽  
pp. 139-156
Author(s):  
Qing-Ming Cheng
Keyword(s):  
Scalar Curvature ◽  
Constant Scalar Curvature ◽  
De Sitter Space ◽  
Second Fundamental Form ◽  
Spacelike Surface ◽  
De Sitter ◽  
Totally Geodesic ◽  
The Second Fundamental Form ◽  
Spacelike Surfaces ◽  
Hyperbolic Cylinder

In this paper, we prove that if M^2 is a complete maximal spacelike surface of an anti-de Sitter space {\bf H}^{4}_{2}(c) with constant scalar curvature, then S=0, S={-10c\over 11}, S={-4c\over 3} or S=-2c, where S is the squared norm of the second fundamental form of M^{2}. Also(1) S=0 if and only if M^2 is the totally geodesic surface {\bf H}^2(c);(2) S={-4c\over 3} if and only if M^2 is the hyperbolic Veronese surface;(3) S=-2c if and only if M^2 is the hyperbolic cylinder of the totally geodesicsurface {\bf H}^{3}_{1}(c) of {\bf H}^{4}_{2}(c).1991 Mathematics Subject Classifaction 53C40, 53C42.

Scalar curvature of Lagrangian Riemannian submersions and their harmonicity

2017 ◽  
Vol 14 (12) ◽  
pp. 1750171 ◽  
Author(s):  
Şemsi Eken Meri̇ç ◽  
Erol Kiliç ◽  
Yasemi̇n Sağiroğlu
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Sufficient Conditions ◽  
Riemannian Submersion ◽  
Hermitian Manifold ◽  
Necessary And Sufficient Conditions ◽  
Riemannian Submersions ◽  
Totally Geodesic ◽  
Totally Umbilical ◽  
Necessary And Sufficient

In this paper, we consider a Lagrangian Riemannian submersion from a Hermitian manifold to a Riemannian manifold and establish some basic inequalities to obtain relationships between the intrinsic and extrinsic invariants for such a submersion. Indeed, using these inequalities, we provide necessary and sufficient conditions for which a Lagrangian Riemannian submersion [Formula: see text] has totally geodesic or totally umbilical fibers. Moreover, we study the harmonicity of Lagrangian Riemannian submersions and obtain a characterization for such submersions to be harmonic.

scholarly journals CR-submanifolds of a locally conformal Kaehler space form

1994 ◽  
Vol 17 (3) ◽  
pp. 511-514
Author(s):  
M. Hasan Shahid
Keyword(s):  
Scalar Curvature ◽  
Sectional Curvature ◽  
Ricci Curvature ◽  
Space Form ◽  
Holomorphic Sectional Curvature ◽  
Totally Geodesic ◽  
Locally Conformal ◽  
Cr Submanifolds

(Bejancu [1,2]) The purpose of this paper is to continue the study ofCR-submanifolds, and in particular of those of a locally conformal Kaehler space form (Matsumoto [3]). Some results on the holomorphic sectional curvature,D-totally geodesic,D1-totally geodesic andD1-minimalCR-submanifolds of locally conformal Kaehler (1.c.k.)-space fromM¯(c)are obtained. We have also discussed Ricci curvature as well as scalar curvature ofCR-submanifolds ofM¯(c).

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