A simple improvement of a differentiable classification result for complete submanifolds

Ezequiel R. Barbosa

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A simple improvement of a differentiable classification result for complete submanifolds

机译：完整子流形的可微分类结果的简单改进

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摘要

We consider M~n, n ≥ 3, an n-dimensional complete sub-manifold of a Riemannian manifold (M~(n+p),g). We prove that if for all point x ∈ M~n the following inequality is satisfied S ≤ 8/3 [K_(min) - 1/4K_(max) +n~2H~2-1, with strictly inequality at one point, where S and H denote the squared norm of the second fundamental form and the mean curvature of M~n respectively, then M~n is either diffeomorphic to a spherical space form or the Euclidean space R~n. In particular, if M~n is simply connected, then M~n is either diffeomorphic to the sphere S~n or the Euclidean space R~n".

机译：我们考虑M〜n，n≥3，这是黎曼流形的n维完整子流形（M〜（n + p），g）。我们证明，如果对于所有点x∈M〜n，满足以下不等式S≤8/3 [K_（min）-1 / 4K_（max）+ n〜2H〜2 / n-1，严格不等式为1点，其中S和H分别表示第二个基本形式的平方范数和M_n的平均曲率，那么M〜n要么是球面空间形式的欧氏空间，要么是欧氏空间R〜n。特别地，如果简单地连接M_n，则M_n或者微分球面S_n或者是欧几里德空间R_n“。

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来源
《Journal of the Mathematical Society of Japan》 |2013年第3期|787-796|共10页
作者
Ezequiel R. Barbosa;
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作者单位

Departamento de Matematica Universidade Federal de Minas Gerais Caixa Postal 702 30123-970, Belo Horizonte MG, Brazil;

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收录信息美国《科学引文索引》(SCI);
原文格式 PDF
正文语种 eng
中图分类
关键词
submanifold; second fundamental form; difrerentiable sphere theorem; Ricci flow; sectional curvature;

机译：子流形第二种基本形式;微分球定理;里奇流截面曲率;

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7. Certain Simple, Unsolvable Problems of Group Theory. V 29,3029Parts I, II, III and IV have appeared in Series A, 57, Nos. 3 and 5; 58, Nos. 2 and 5 of these Proceedings, as well as Indag. Math., 16, Nos. 3 and 5 (1954); 17, Nos. 2 and 5 (1955).In Part I, page 234, the ninth and tenth lines following the displayed material, for both occurrences of A read D and for both occurrences of B read E: In part II, page 497, for all occurrences of M in Diagrams A and D read C. In Part IV, page 574, the third displayed line, for zαiLread z―αiL. 19).30The proof of the unsolvability of the word problem contained in this and subsequent paper is not contained in the dissertation Several simple unsolvable problems of group theory related to the word problem. (See footnote 1 of Part I.) Our proof was finally completed during the period 1954-56 while in residence at the Institute for Advanced Study. We were supported in 1954-55 directly by the Institute and in 1955-56 by National Science Foundation contract G-1974. Certain improvements were evolved ans preparations for publication were completed while the author held a Fulbright grant to the University of Oslo.Our thanks are due to Professor Kurt Gödel for his kind encouragement in these matter; certain related problems for study he has suggested we hope to deal with later. When an earlier version of Lemmas 30 and 31 was explained at a colloquium at the University of Michigan, August 3, 1956, Professor Roger Lyndon suggested an improvement which is incorporated in our present version. We are indeed indebted to Dr. John Addison and Dr. Michael Rabin for checking many of the new details during July of 1956.) [O] . Boone William W. 1957

机译：群体理论的某些简单，不可解决的问题。 V 29,3029 I，II，III和IV部分出现在A系列57、3和5号中； 58，第2和5号程序，以及Indag。 Math。，16，Nos 3 and 5（1954）; 》，第17卷，第2期和第5期（1955年）。在第一部分，第234页中，显示的材料之后的第九行和第十行，出现A读D和出现B都读E：在第二部分，第497页，对于图A和D中所有出现的M，请读C。在第Ⅳ部分，第574页中，第三行显示为zαiLreadz-αiL。 19）.30本论文及后续论文中所包含的单词问题的不可解性的证明不包含在本文中。（请参阅第I部分的脚注1。）我们的证明终于在1954-56年期间居住在高级研究所期间完成了。该研究所在1954-55年间直接为我们提供了支持，在1955-56年间得到了美国国家科学基金会的G-1974合同的支持。在作者获得奥斯陆大学富布赖特奖学金的同时，对出版物的改进和准备工作也有所进展。感谢库尔特·哥德尔教授在此方面的大力鼓励。他建议我们研究一些相关的问题，希望以后再解决。 1956年8月3日，在密歇根大学的一个座谈会上解释了Lemmas 30和31的早期版本时，Roger Lyndon教授建议进行改进，并将其纳入当前版本。我们确实要感谢John Addison博士和Michael Rabin博士在1956年7月检查了许多新细节。）

A simple improvement of a differentiable classification result for complete submanifolds

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