Displaying similar documents to “Conformally flat submanifolds”

Conformal nullity of isotropic submanifolds

Vladimir Rovenski (2005)

Annales Polonici Mathematici

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We introduce and study submanifolds with extrinsic curvature and second fundamental form related by an inequality that holds for isotropic submanifolds and becomes equality for totally umbilical submanifolds. The dimension of umbilical subspaces and the index of conformal nullity of these submanifolds with low codimension are estimated from below. The corollaries are characterizations of extrinsic spheres in Riemannian spaces of positive curvature.

Semiparallel isometric immersions of 3-dimensional semisymmetric Riemannian manifolds

Ülo Lumiste (2003)

Czechoslovak Mathematical Journal

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A Riemannian manifold is said to be semisymmetric if R ( X , Y ) · R = 0 . A submanifold of Euclidean space which satisfies R ¯ ( X , Y ) · h = 0 is called semiparallel. It is known that semiparallel submanifolds are intrinsically semisymmetric. But can every semisymmetric manifold be immersed isometrically as a semiparallel submanifold? This problem has been solved up to now only for the dimension 2, when the answer is affirmative for the positive Gaussian curvature. Among semisymmetric manifolds a special role is played...

Willmore submanifolds in the unit sphere.

Guo Zhen (2004)

Collectanea Mathematica

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In this paper we generalize the self-adjoint differential operator (used by Cheng-Yau) on hypersurfaces of a constant curvature manifold to general submanifolds. The generalized operator is no longer self-adjoint. However we present its adjoint operator. By using this operator we get the pinching theorem on Willmore submanifolds which is analogous to the pinching theorem on minimal submanifold of a sphere given by Simon and Chern-Do Carmo-Kobayashi.