Positive scalar curvature and the Dirac operator on complete riemannian manifolds
Positive scalar curvature and the Dirac operator on complete riemannian manifolds
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DOI：
10.1007/bf02953774
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发表时间：
1983-12
期刊：
影响因子：
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通讯作者：
Mikhael Gromov;Mikhael Gromov;H. B. Lawson;H. B. Lawson
中科院分区：
文献类型：
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作者：
Mikhael Gromov;Mikhael Gromov;H. B. Lawson;H. B. Lawson
这项研究的主要目的之一是理解具有正标量曲率度量的空间。近年来，这一主题一直是活跃研究的焦点，我们想首先简要讨论一下这方面的进展(1)。当然，黎曼流形最简单的不变量之一就是它的标量曲率函数。在一般维度中，该函数(其在一点的值正好是平均截面曲率)是局部几何的弱度量，人们可能会怀疑它与流形的全局拓扑无关。这种怀疑的证据可以在卡兹丹-华纳和奥宾的作品中找到。例如，在[KW]中，证明了在维度为^3的紧致流形上，每个在某处为负的光滑函数都是某个黎曼度量的标量曲率。然而，有趣的事实是，存在高维流形，它们没有度量，其标量曲率处处都是^o。这类流形的第一个例子是由A.Lichnerowicz[Li]在1962年给出的，他的推理如下。在任何黎曼自旋流形上都存在一个基本的椭圆算子，称为狄拉克算子(2)。利用Bochner的方法，Richnerowicz证明了在具有正数量曲率的紧致流形上，该算子是可逆的。在4维空间中，他通过Atiyah-Singer指数定理得出结论，流形的一个基本拓扑不变量，称为A-亏格，必须为零。对于任意定向的4-流形M，8A(M)==Signature(M)是一个事实。由于存在许多非零签名的自旋4-流形，而且A是(<；乘法)不变量，上述论点产生了许多不具有正标量曲率的流形的例子。
One of the principal aims of this study is to understand spaces which carry metrics of positive scalar curvature. In recent years this subject has been the focus of lively research, and we would like to begin with a brief discussion of the developments (1). Certainly, one of the simplest invariants of a riemannian manifold is its scalar curvature function. In general dimensions, this function (whose value at a point is just the average sectional curvature) is a weak measure of the local geometry, and one might suspect it to be unrelated to the global topology of the manifold. Evidence for this suspicion can be found in the work ofKazdan-Warner and Aubin. For example, in [KW] it is proved that on a compact manifold of dimension^ 3, every smooth function which is negative somewhere, is the scalar curvature of some riemannian metric. However, the intriguing fact is that there are manifolds of high dimension which carry no metrics whose scalar curvature is everywhere^ o.The first examples of such manifolds were given in 1962 by A. Lichnerowicz [Li], who reasoned as follows. Over any riemannian spin manifold there exists a fundamental elliptic operator, called the Dirac operator (2). Using Bochner's method, Lichnerowicz showed that on compact manifolds of positive scalar curvature, this operator is invertible. In dimensions 4^ he then concluded, via the Atiyah-Singer Index Theorem, that a certain basic topological invariant of the manifold, called the A-genus, must vanish. For any oriented 4-manifold M, it is a fact that 8A (M)== signature (M). Since there are many spin 4-manifolds of non-zero signature, and since A is a (< multiplicative" invariant, the above argument produces many examples of manifolds which do not carry positive scalar curvature.