Some implications of the generalized Gauss-Bonnet theorem
Some implications of the generalized Gauss-Bonnet theorem
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广义高斯-博内定理的一些含义
DOI：
10.1090/s0002-9947-1964-0163271-8
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发表时间：
1964
影响因子：
1.3
通讯作者：
S. Goldberg
中科院分区：
文献类型：
--
作者：
R. Bishop;S. Goldberg
1. 简介。也许微分几何最重要的方面是处理黎曼流形 M 的曲率性质与其拓扑结构之间的关系。与此相关的漂亮结果之一是（广义）高斯-邦内定理，该定理将紧致定向偶维流形的曲率与重要的拓扑不变量（即 M 的欧拉-庞加莱特征 x(M)）联系起来。在二维情况下，高斯曲率的符号决定了 x(M) 的符号。此外，如果高斯曲率同样消失，#(M) 也同样消失。在更高维度中，Gauss-Bonnet 公式（参见第 3 节）并不那么简单，并且引出了以下重要问题。偶数维 n = 2m 的紧致有向黎曼流形，其截面曲率均为非负，是否具有非负欧拉-庞加莱特性，并且 ij 截面曲率非正，则 (-l)m*(M) = 0？
1. Introduction. Perhaps the most significant aspect of differential geometry is that which deals with the relationship between the curvature properties of a Riemannian manifold M and its topological structure. One of the beautiful results in this connection is the (generalized) Gauss-Bonnet theorem which relates the curvature of compact and oriented even-dimensional manifolds with an important topological invariant, viz., the Euler-Poincar6 characteristic x(M) of M. In the 2-dimensional case, the sign of the Gaussian curvature determines the sign of x(M). Moreover, if the Gaussian curvature vanishes identically, so does #(M). In higher dimensions, the Gauss-Bonnet formula (cf. §3) is not so simple, and one is led to the following important Question. Does a compact and oriented Riemannian manifold of even dimension n = 2m whose sectional curvatures are all non-negative have non-negative Euler-Poincare characteristic, and ij the sectional curvatures are nonpositive is (-l)m*(M) = 0?