|
|
| |
| SUBMANIFOLDS OF A HIGHER DIMENSIONAL SPHERE |
| |
Citation： |
Huang Xuanguo.SUBMANIFOLDS OF A HIGHER DIMENSIONAL SPHERE[J].Chinese Annals of Mathematics B,1983,4(1):33~40 |
| Page view： 736
Net amount： 549 |
Authors： |
Huang Xuanguo; |
|
|
| Abstract： |
Let M be an m-dimensional manifold immersed in $\[{S^{m + k}}(r)\]$. Then $\[\Delta X = \mu H - \frac{m}{{{r^2}}}X\]$.
where X is the position vector of M and H is a unit normal vector field which is orthogonal to X everywhere.
If M is a compact connected manifold with parallel mean curvature vector field $\[\xi \]$ immersed in $\[{S^{m + k}}(r)\]$, and the sectional curvature of M is not less than $\[\frac{1}{2}(\frac{1}{{{r^2}}} + {\left| \xi \right|^2})\]$, then M is a small sphere.
For a compact connected hypersurface M in $\[{S^{m + 1}}(r)\]$, if the sectional curvature is nonnegative and the scalar curvature is proportional to the mean curvature everywhere, then M is a totally umbilical hypersurface or the multiplication of two totally umbilical submanifolds. |
Keywords： |
|
Classification： |
|
|
|
Download PDF Full-Text
|
|
|
|
