Vector fields and Ricci curvature
Vector fields and Ricci curvature
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DOI：
10.1090/s0002-9904-1946-08647-4
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发表时间：
1946-09
影响因子：
1.3
通讯作者：
S. Bochner
中科院分区：
文献类型：
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作者：
S. Bochner
我们将证明在具有正定黎曼度量的紧致流形上某些类型的向量场的不存在性定理，其Ricci曲率是处处正的或处处负的。实际上，我们对曲率和紧性的要求都有一些放宽。我们将讨论具有通常度量的真实的空间和具有厄米度量的复解析空间。在后一种情况下，我们将对度规施加一定的限制，这首先由E明确地说明。凯勒，这对我们的论证是不可或缺的。为了阐明这种限制的作用，我们将系统地介绍厄米度规理论。对于正曲率，我们将得到这样的定理：在紧致空间上不存在散度和旋度都为零的向量场。在复情况下，不存在任何向量场，其协变分量是复参数的解析函数。如果我们只假设曲率是非负的，那么在空间平坦的方向上会有一些“例外”的向量场。一个主要结果将是以下定理亚纯函数。如果一个具有正曲率的复空间被有限个邻域覆盖，如果在每个邻域中定义一个亚纯函数元，并且如果任何两个亚纯元的差在任何亚纯元重叠的地方都是全纯的，则存在一个亚纯函数与每个给定的亚纯元相差一个全纯函数。在以前的一篇论文中，
We shall prove theorems on nonexistence of certain types of vector fields on a compact manifold with a positive definite Riemannian metric whose Ricci curvature is either everywhere positive or everywhere negative. Actually we shall have some relaxations of the requirements both as to curvature and as to compactness. We shall deal with real spaces with a customary metric and with complex analytic spaces with an Hermitian metric. In the latter case we shall impose on the metric a certain restriction, first explicitly stated by E. Kaehler, which will be quite indispensable to our argument. In order to elucidate the rôle of this restriction we shall include a systematic introduction to the theory of Hermitian metric. For positive curvature we shall have the theorem that on a compact space there exists no vector field for which the divergence and curl both vanish. In the complex case there exists no vector field whatsoever whose covariant components are analytic functions in the complex parameters. If we only assume that the curvature is nonnegative, then there are some "exceptional" vector fields in directions of spatial flatness. A principal result will be the following theorem on meromorphic functions. If a complex space with positive curvature is covered by a finite number of neighborhoods, if a meromorphic functional element is defined in each neighborhood, and if the difference of any two meromorphic elements is holomorphic wherever the elements overlap, then there exists one meromorphic function on the space which differs by a holomorphic function from each meromorphic element given. In a previous paper this conclusion was drawn in the