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Please use this identifier to cite or link to this item: http://hdl.handle.net/123456789/10875

Title: 3-Dimensional minimal submanifolds of a sphere
Authors: Deshmukh, Sharief
Ghazal, Tahsin
Keywords: -dimensional minimal submanifold
Issue Date: 1992
Publisher: Soochow Journal of Mthematics ,
Citation: Vol. 18 No.2, 1992, 159-162.
Abstract: Let $M$ denote a 3-dimensional minimal submanifold of the unit $n$-sphere $S\sp n(1)$. The authors present two results that express a connection between the diffeomorphism class of $M$ and the “size” of the second fundamental form $h$ of $M$. Theorem 1. Let $M$ be a 3-dimensional minimal submanifold of $S\sp n(1)$. If the second fundamental form of $M$ satisfies $\Vert h(X,X)\Vert \sp 2<\frac12$ for each unit vector field $X\in{\scr X}(M)$, then $M$ is diffeomorphic to one of the following: (i) ${\bf R}{\rm P}\sp 3$, (ii) a 3-dimensional lens space, or (iii) $S\sp 3$. Theorem 2. Let $M$ be a compact 3-dimensional minimal submanifold of $S\sp n(1)$. If the square $S$ of the length of the second fundamental form satisfies $S<2$, then $M$ is diffeomorphic to one of the following: (i) ${\bf R}{\rm P}\sp 3$, (ii) a 3-dimensional lens space, or (iii) $S\sp 3$.
URI: http://hdl.handle.net/123456789/10875
Appears in Collections:College of Science

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