FINITENESS THEOREMS FOR RIEMANNIAN MANIFOLDS.
FINITENESS THEOREMS FOR RIEMANNIAN MANIFOLDS.
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DOI：
10.2307/2373498
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发表时间：
1970
影响因子：
1.7
通讯作者：
J. Cheeger
中科院分区：
文献类型：
--
作者：
J. Cheeger
1.本文的目的是证明，如果对与黎曼度量有关的某些几何量的大小给出任意固定的界，则允许一个度量满足这些界的紧致n维流形的自同构类的集合是有限的.作为一个应用，我们证明了紧致n-流形（n，4）中某些特征数为非零且允许非负曲率的爱因斯坦度量的自同态类的集合是有限的.我们的主要工具是一个定理，给出了指数映射的内射半径的下界。这些结果代表了作者博士论文的一部分的改进版本，他再次感谢S。感谢Bochner和J. Simons的建议和鼓励。推论3的C部分的一个较弱的版本。3独立于A。
1. The purpose of this paper is to show that if one puts arbitrary fixed bounds on the size of certain geometrical quantities associated with a riemannian metric, then the set of diffeomorphism classes of compact ndimensional manifolds admitting a metric for which these bounds are satisfied is finite. As an application, we show that the set of diffeomorphism classes of compact n-manifolds, (n , 4) for which some characteristic number is nonzero, and which admit an Einstein metric of nonnegative curvature, is finite. Our main tool is a theorem giving a lower bound for the injectivity radius of the exponential map. The results represent an improved version of a portion of the author's doctoral thesis and he again wishes to thank Professors S. Bochner and J. Simons for their advice and encouragement. A weaker version of part C) of Corollary 3. 3 is due independently to A.