Minimal Submanifolds of a Sphere with Second Fundamental Form of Constant Length

Chern, Shiing-Shen; Carmo, Manfredo do; Kobayashi, Shimpei

doi:10.1007/978-3-642-48272-4_2

Cited by 121 publications

(81 citation statements)

References 2 publications

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“…There are many studies on compact hypersurfaces in an (n + 1)-dimensional unit sphere S n+1 (1) (see [1], [2], [3], [5]- [12]). In this paper, let us also denote by M a compact hypersurfaces in S n+1 (1).…”

Section: Preliminariesmentioning

confidence: 99%

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Linear Weingarten Hypersurfaces in a Unit Sphere

Li¹,

Suh²,

Wei³

2009

Bulletin of the Korean Mathematical Society

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Section: Preliminariesmentioning

confidence: 99%

“…If H = 0 and |B| 2 = 2, we know that M = S 1 (c) × S 1 ( √ 1 − c 2 ) from a result of [5]. If |B| 2 − 2H 2 = 0, M is totally umbilical.…”

Section: Proofs Of Theorem 13 and Theorem 14mentioning

confidence: 99%

Linear Weingarten Hypersurfaces in a Unit Sphere

Li¹,

Suh²,

Wei³

2009

Bulletin of the Korean Mathematical Society

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“…. , n + 2 and so, using lemma 1 we conclude that (n − |σ| 2 ) f, a i = 0 for any i, which is possible only if n − |σ| 2 = 0 on M. Now the result of Chern, do Carmo and Kobayashi [5] says that M is locally congruent to a Clifford minimal hypersurface. Thus M is congruent either to the Clifford hypersurface S n 1 (R 1 ) × S n 2 (R 2 ) ⊂ S n+1 (with n 1 R 2 2 = n 2 R 2 1 ) itself or to a nontrivial finite covering of it.…”

Section: Theorem 3 the Only Compact Two-sided Minimal Hypersurfaces mentioning

confidence: 84%

Compact minimal hypersurfaces with index one in the real projective space

Carmo¹,

Ritoré²,

Ros³

2000

Comment. Math. Helv.

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Abstract. Let M n be a compact (two-sided) minimal hypersurface in a Riemannian manifold M n+1 . It is a simple fact that if M has positive Ricci curvature then M cannot be stable (i. e. its Jacobi operator L has index at least one). If M = S n+1 is the unit sphere and L has index one, then it is known that M must be a totally geodesic equator.We prove that if M is the real projective space P n+1 = S n+1 /{±}, obtained as a metric quotient of the unit sphere, and the Jacobi operator of M has index one, then M is either a totally geodesic sphere or the quotient to the projective space of the hypersurface S n 1 (R 1 ) × S n 2 (R 2 ) ⊂ S n+1 obtained as the product of two spheres of dimensions n 1 , n 2 and radius R 1 , R 2 , with n 1 + n 2 = n, R 2 1 + R 2 2 = 1 and n 1 R 2 2 = n 2 R 2 1 .

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“…From now on, we agree on the following index ranges: 3 , e 1 * , e 2 * , e 3 * } to be a local orthonormal frame field of the tangent bundle TS 6 such that e i lies in TM and e i * = Je i lies in NM. Let {ω 1 , ω 2 , ω 3 , ω 1 * , ω 2 * , ω 3 * } be the associated coframe field.…”

Section: G(x X) = 0; G(x Y ) + G(y X) = 0; G(x Jy ) + Jg(x Y ) =mentioning

confidence: 99%

Classification of Lagrangian Willmore Submanifolds of the Nearly Kaehler 6-Sphere $S^6(1)$ With Constant Scalar Curvature

Li¹,

Wei²

2006

Glasgow Math. J.

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Abstract. In this paper, we classify 3-dimensional Lagrangian Willmore submanifolds of the nearly kaehler 6-sphere S 6 (1) with constant scalar curvature.2000 Mathematics Subject Classification. 53 B 25; 53 D 12; 53 C 42. Introduction.It is well known that a 6-dimensional sphere S 6 (1) admits an almost kaehler structure J by making use of the Cayley system. Many interesting theorems about the topology and the geometry of nearly kaehler manifolds have been proved (see [2,4,7]). There have been many results on geometry of submanifolds in a kaehler manifold. Especially, submanifolds (called Lagrangian submanifolds) for which J interchanges the tangent and normal spaces. The theory of Lagrangian submanifolds in a nearly kaehler manifold was studied by many authors (cf. e.g. N. Ejiri, B. Y. Chen, F. Dillen, L. Vrancken and L. Verstraelen etc.). About Lagrangian submanifolds of S 6 (1), in [5], the authors classified the compact Lagrangian submanifolds of S 6 (1) whose sectional curvatures satisfy K ≥ 1 16 . In [2], the authors classified the Lagrangian submanifolds of S 6 (1) with constant scalar curvature that realize the Chen's inequality. In this paper, we classify Lagrangian Willmore submanifold of the nearly kaehler 6-sphere S 6 (1) with constant scalar curvature and obtain all possible values for the norm square of the second fundamental form S about these submanifolds. It is similar to Chern's conjecture which states that the set of all possible values for S of a compact minimal submanifold in the sphere with S = constant is a limit set.

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Minimal Submanifolds of a Sphere with Second Fundamental Form of Constant Length

Cited by 121 publications

References 2 publications

Linear Weingarten Hypersurfaces in a Unit Sphere

Linear Weingarten Hypersurfaces in a Unit Sphere

Compact minimal hypersurfaces with index one in the real projective space

Classification of Lagrangian Willmore Submanifolds of the Nearly Kaehler 6-Sphere $S^6(1)$ With Constant Scalar Curvature

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