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| ON THE TOPOLOGY, VOLUME, DIAMETER AND GAUSS MAP IMAGE OF SUBMANIFOLDS IN A SPHERE |
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Citation： |
WU Bingye.ON THE TOPOLOGY, VOLUME, DIAMETER AND GAUSS MAP IMAGE OF SUBMANIFOLDS IN A SPHERE[J].Chinese Annals of Mathematics B,2004,25(2):207~212 |
| Page view： 1057
Net amount： 804 |
Authors： |
WU Bingye; |
Foundation： |
Project supported by the Fund of the Education Department of Zhejiang Province of China
(No.20030707). |
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| Abstract： |
In this paper, the author uses Gauss map to study the topology,
volume and diameter of submanifolds in a sphere. It is proved that
if there exist $\varepsilon,\ 1\ge\varepsilon>0$ and a fixed unit
simple $p$-vector $a$ such that the Gauss map $g$ of an
$n$-dimensional complete and connected submanifold $M$ in
$S^{n+p}$ satisfies $\langle g,a\rangle\ge\varepsilon$, then $M$
is diffeomorphic to $S^n$, and the volume and diameter of $M$
satisfy $\varepsilon^n$vol$(S^n)\le$vol$(M)\le$
vol$(S^n)/\varepsilon$ and
$\varepsilon\pi\le$diam$(M)\le\pi/\varepsilon$, respectively. The
author also characterizes the case where these inequalities become
equalities. As an application, a differential sphere theorem for
compact submanifolds in a sphere is obtained. |
Keywords： |
Gauss map, Volume, Diameter, Differential sphere theorem |
Classification： |
53C42, 53B30 |
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