|
|
| |
| MINIMAL SUBMANIFOLDS IN A RIEMANNIAN MANIFOLD OF QUASI CONSTANT CURVATURE |
| |
Citation： |
Bai zhengguo.MINIMAL SUBMANIFOLDS IN A RIEMANNIAN MANIFOLD OF QUASI CONSTANT CURVATURE[J].Chinese Annals of Mathematics B,1988,9(1):32~37 |
| Page view： 872
Net amount： 662 |
Authors： |
Bai zhengguo; |
Foundation： |
Projects supported by the Science Fund of the Chinese Academy of Sciences. |
|
|
| Abstract： |
A Riemannian manifold $\[{V^m}\]$ which admits isometric imbedding into two spaces $\[{V^{m + p}}\]$ of different constant curvatures is called a manifold of quasi constant $\[{\rm{curvatur}}{{\rm{e}}^{[2]}}\]$. The Riemannian curvature of $\[{V^m}\]$ is expressible in the form
$$\[{K_{ABCD}} = a({g_{AC}}{g_{BD}} - {g_{AD}}{g_{BC}}) + b({g_{AC}}{\lambda _B}{\lambda _D} + {g_{BD}}{\lambda _A}{\lambda _C} - {g_{AD}}{\lambda _B}{\lambda _C} - {g_{BC}}{\lambda _A}{\lambda _D}),(1 = \sum\limits_{}^{} {{g_{AB}}{\lambda _A}{\lambda _B}} )\]$$
and conversely. In this paper it is proved that if $\[{M^n}\]$ is any compact minimal submanifold without boundary in a Riemannian manifold $\[{V^{n + p}}\]$ of quasi constant curvature, then
$$\[{{{\{ \left( {2 - \frac{1}{p}} \right){\sigma ^2} - [na + \frac{1}{2}(b - \left| b \right|)(n + 1)]\sigma + n(n - 1){b^2}\} }^*}| \ge 0}\]$$
where $\[\sigma \]$ is the square of the norm of the second fundamental form of $\[{M^n}\]$. When $\[{V^{n + p}}\]$ is a
manifold of constant curvature, $\[b = 0\]$, the above inequality reduces to that of Simons. |
Keywords： |
|
Classification： |
|
|
|
Download PDF Full-Text
|
