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Displaying similar documents to “Global pinching theorems of submanifolds in spheres.”

On total curvature of immersions and minimal submanifolds of spheres

Giovanni Rotondaro (1993)

Commentationes Mathematicae Universitatis Carolinae

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For closed immersed submanifolds of Euclidean spaces, we prove that ∫ | μ | 2 d V ≥ V / R 2 , where μ is the mean curvature field, V the volume of the given submanifold and R is the radius of the smallest sphere enclosing the submanifold. Moreover, we prove that the equality holds only for minimal submanifolds of this sphere.

A pinching theorem on complete submanifolds with parallel mean curvature vectors

Ziqi Sun (2003)

Colloquium Mathematicae

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Let M be an n-dimensional complete immersed submanifold with parallel mean curvature vectors in an (n+p)-dimensional Riemannian manifold N of constant curvature c > 0. Denote the square of length and the length of the trace of the second fundamental tensor of M by S and H, respectively. We prove that if S ≤ 1/(n-1) H² + 2c, n ≥ 4, or S ≤ 1/2 H² + min(2,(3p-3)/(2p-3))c, n = 3, then M is umbilical. This result generalizes the...

A pointwise inequality in submanifold theory

P. J. De Smet, F. Dillen, Leopold C. A. Verstraelen, L. Vrancken (1999)

Archivum Mathematicum

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We obtain a pointwise inequality valid for all submanifolds M n of all real space forms N n + 2 ( c ) with n ≥ 2 and with codimension two, relating its main scalar invariants, namely, its scalar curvature from the intrinsic geometry of M n , and its squared mean curvature and its scalar normal curvature from the extrinsic geometry of M n in N m ( c ) .