OPTIMAL INEQUALITIES FOR MULTIPLY WARPED PRODUCT SUBMANIFOLDS
OPTIMAL INEQUALITIES FOR MULTIPLY WARPED PRODUCT SUBMANIFOLDS
复制标题
多重扭曲产品子流形的最优不等式
DOI：
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
F. Dillen
中科院分区：
文献类型：
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作者：
Bang‐Yen Chen;F. Dillen
在较早的论文 (3) 中，第一作者证明，对于将扭曲乘积 N1 £f N2 任意等距浸入截面曲率恒定 c 的黎曼 m 流形中，扭曲函数 f 满足最优一般不等式： ∫ f f • (n1 + n2)2 4n2 H 2 + n1c；其中ni = dimNi，i = 1;2，H 2 是平方平均曲率， 是N1 的拉普拉斯算子。此外，他在（2）中证明了对于 CR 扭曲产品 NT £f N？在凯勒流形中，第二基本形式 h 满足 jjhjj 2 ‚ 2n2jjr(lnf)jj 2 ，其中 n2 = dimN?。在本文中，我们将这些不等式扩展到任意黎曼或凯勒流形中的扭曲乘积流形的乘法。我们还提供了一些例子来说明我们的结果是清晰的。此外，还获得了多个应用。
In an earlier paper (3) the flrst author proved that, for any isomet- ric immersion of a warped product N1 £f N2 into a Riemannian m-manifold of constant sectional curvature c, the warping function f satisfles the optimal general inequality: ¢f f • (n1 + n2)2 4n2 H 2 + n1c; where ni = dimNi, i = 1;2, H 2 is the squared mean curvature, and ¢ is the Laplacian operator of N1. Moreover, he proved in (2) that for a CR-warped product NT £f N? in a Kaehler manifold, the second fundamental form h satisfles jjhjj 2 ‚ 2n2jjr(lnf)jj 2 , where n2 = dimN?. In this article we extend these inequalities to multiply warped product manifolds in an arbitrary Rie- mannian or Kaehlerian manifold. We also provide some examples to illustrate that our results are sharp. Moreover, several applications are also obtained.