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).

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).

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 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.

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 PINCHING THEOREMS FOR TOTALLY REAL MINIMAL SUBMANIFOLDS INCPn

2009 ◽  
Vol 51 (2) ◽  
pp. 331-339
Author(s):  
CENGİZHAN MURATHAN ◽  
CİHAN ÖZGÜR
Keyword(s):  
Projective Space ◽  
Scalar Curvature ◽  
Minimal Submanifolds ◽  
Minimal Submanifold ◽  
Real Projective Space ◽  
Clifford Torus ◽  
Open Part ◽  
The Real ◽  
Totally Real

AbstractLetMbe ann-dimensional totally real minimal submanifold inCPn. We prove that ifMis semi-parallel and the scalar curvature τ,$\frac{-(n-1)(n-2)(n+1)}{2}\leq \tau \leq 0$, thenMis an open part of the Clifford torusTn⊂CPn. IfMis semi-parallel and the scalar curvature τ,$n(n-1)\leq \tau \leq \frac{n^{3}-3n+2}{2}$, thenMis an open part of the real projective spaceRPn.

Compact hypersurfaces in a unit sphere

2005 ◽  
Vol 135 (6) ◽  
pp. 1129-1137 ◽  
Author(s):  
Qing-Ming Cheng ◽  
Shichang Shu ◽  
Young Jin Suh
Keyword(s):  
Scalar Curvature ◽  
Unit Sphere ◽  
Mean Curvature ◽  
Partial Answer ◽  
Principal Curvatures ◽  
Riemannian Product ◽  
Compact Hypersurface ◽  
Image Position ◽  
Zero Mean Curvature

We study curvature structures of compact hypersurfaces in the unit sphere Sn+1(1) with two distinct principal curvatures. First of all, we prove that the Riemannian product is the only compact hypersurface in Sn+1(1) with two distinct principal curvatures, one of which is simple and satisfies where n(n − 1)r is the scalar curvature of hypersurfaces and c2 = (n − 2)/nr. This generalized the result of Cheng, where the scalar curvature is constant is assumed. Secondly, we prove that the Riemannian product is the only compact hypersurface with non-zero mean curvature in Sn+1(1) with two distinct principal curvatures, one of which is simple and satisfies This gives a partial answer for the problem proposed by Cheng.

On Lagrangian Catenoids

2007 ◽  
Vol 50 (3) ◽  
pp. 321-333 ◽  
Author(s):  
David E. Blair
Keyword(s):  
General Problem ◽  
Lagrangian Submanifolds ◽  
Minimal Submanifold ◽  
Conformally Flat ◽  
Totally Geodesic ◽  
Schouten Tensor ◽  
Multiplicity One

AbstractRecently I. Castro and F.Urbano introduced the Lagrangian catenoid. Topologically, it is ℝ × Sn–1 and its induced metric is conformally flat, but not cylindrical. Their result is that if a Lagrangian minimal submanifold in ℂn is foliated by round (n – 1)-spheres, it is congruent to a Lagrangian catenoid. Here we study the question of conformally flat, minimal, Lagrangian submanifolds in ℂn. The general problem is formidable, but we first show that such a submanifold resembles a Lagrangian catenoid in that its Schouten tensor has an eigenvalue of multiplicity one. Then, restricting to the case of at most two eigenvalues, we show that the submanifold is either flat and totally geodesic or is homothetic to (a piece of) the Lagrangian catenoid.

scholarly journals Examples and classification of Riemannian submersions satisfying a basic equality

2005 ◽  
Vol 72 (3) ◽  
pp. 391-402 ◽  
Author(s):  
Bang-Yen Chen
Keyword(s):  
Riemannian Manifold ◽  
Unit Sphere ◽  
Isometric Immersion ◽  
Riemannian Submersion ◽  
Sharp Inequality ◽  
Riemannian Submersions ◽  
Equality Case ◽  
Totally Geodesic ◽  
Basic Equality

In an earlier article we obtain a sharp inequality for an arbitrary isometric immersion from a Riemannian manifold admitting a Riemannian submersion with totally geodesic fibres into a unit sphere. In this article we investigate the immersions which satisfy the equality case of the inequality. As a by-product, we discover a new characterisation of Cartan hypersurface in S4.

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