Scalar curvature of spheres
Scalar curvature of spheres
复制标题
球体的标量曲率
DOI：
10.1007/bf01230287
复制
发表时间：
1989
影响因子：
0.8
通讯作者：
Osamu Kobayashi
中科院分区：
文献类型：
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作者：
Osamu Kobayashi
已知，如果紧致n流形M， n~ 3，允许正标量曲率的度量，则任意M 1s的光滑函数可实现为M的某个度量的标量曲率函数10n （cf.[lJ]）。这篇论文试图证明，即使我们假设公制的总体积为单位，这一表述也是成立的。在之前的论文[2]中，除了正常数函数之外，都解决了这个问题。因此我们只需要找到单位体积和标量曲率等于任意给定正常数的度规。一个困难是我们不能应用Yamabe问题，因为它只提供了小于或等于标准球体的常数标量曲率~当体积1s归一化时。另一方面，在一些明显的情况下，我们可以很容易地在体积约束下得到任何正常数标量曲率。也就是说，当M是积流形MtX M时
It is known that, if a compact n-manifold M, n~ 3, admits ametrie of positive scalar curvature, then any smooth function of M 1s realized as the scalar curvature funct10n of some metric of M (cf.[lJ). Th1s paper 1s an attempt to show th1s statement will be true'even 1f we assume the metric has unit total volume. In the previous paper [2], this problem was solved except for positive constant funct1ons. Therefore we have only to find metrics with unit velume and with scalar curvature equal to arbitrarily given positive constant. One difficulty is that we cannot apply the Yamabe problem because it provides only constant scalar curvature less than or equal te that of the standard sphere~ hen the volume 1s normalized. On the other hand there are obvious cases in wh1ch we can easily get any positive constant scalar curvature under the volume constraint. That is, when M is a produet manifold MtX M