Möbius Characterization of Some Submanifolds in the Unit Sphere

• DOI: 10.1007/s00025-006-0221-5
• Publication Date: 2006-12-05T16:09:15Z
• Authors: Zhen Guo; Kaiwen Guo
• Keywords: Characterization; Codimension; Unit sphere

Let Mm be an m-dimensional umbilic-free submanifold in the m + p-dimensional unit sphere Sm+p. Three basic invariants of Mm under the Mobius transformation group of Sm are a 1-form Φ, called Mobius form, a symmetric (0,2) tensor A, called Blaschke tensor, and a positive definite (0,2) tensor g, called Mobius metric. We denote the Mobius scalar curvature by R and the trace-free Blaschke tensor by \( \tilde{{\bf A}}: = {\bf A} - \frac{1}{m}\hbox{tr}({{\bf A}}){\bf {g}} \). In this paper, we prove a local classification result under the assumption of vanishing Mobius form and an inequality of the type $$ c_{1} \left\| {\left. \tilde{{\bf A}}\right\| \le R - c_{2} } \right. , $$ where c1, c2 are appropriate real constants, c1 depending on the dimension, and c2 on the dimension and codimension.