A simple improvement of a differentiable classification result for complete submanifolds

作者：Barbosa Ezequiel R^*

来源：Journal of the Mathematical Society of Japan, 2013, 65(3): 787-796.

DOI：10.2969/jmsj/06530787

• submanifold
• second fundamental form
• differentiable sphere theorem
• Ricci flow
• sectional curvature

We consider M-n, n %26gt;= 3, an n-dimensional complete submanifold of a Riemannian manifold ((M) over bar (n+P),(g) over bar). We prove that if for all point x is an element of M-n the following inequality is satisfied %26lt;br%26gt;S %26lt;= 8/3 ((K) over bar (min) -1/4 (K) over bar (max)) + n(2)H(2)/n - 1, %26lt;br%26gt;with strictly inequality at one point, where S and H denote the squared norm of the second fundamental form and the mean curvature of M-n respectively, then M-n is either diffeomorphic to a spherical space form or the Euclidean space R-n. In particular, if M-n is simply connected, then M-n is either diffeomorphic to the sphere Sn or the Euclidean space R-n.

• 出版日期2013-7