A direct approach to the determination of Gaussian and scalar curvature functions
A direct approach to the determination of Gaussian and scalar curvature functions
复制标题
确定高斯曲率函数和标量曲率函数的直接方法
DOI：
10.1007/bf01425558
复制
发表时间：
1975
影响因子：
3.1
通讯作者：
F. W. Warner
中科院分区：
文献类型：
--
作者：
J. Kazdan;F. W. Warner
在[9]中，我们精确地描述了紧致连通2-流形M上的高斯曲率函数，并给出了高维情况下标量曲率的相关信息。我们建议读者参考[6，8，9]的介绍，以获得历史和书目信息。在这里，我们想提出一个稍微不同的证明上述结果。虽然证明的基本轮廓与[9]中的几乎相同，但不同之处在于它采用了Fischer-Marsden [4]的更直接的方法，并且根本没有使用保形变形。这种直接方法在技术上和概念上都比较简单，应该更容易为微分几何学家所接受。另一方面，有一个惩罚，即没有获得我们以前工作的共形信息[9]，特别是关于逐点共形变形的更深更精细的信息[6，8，12]。给定M上的度量g，可以通过包含度量的一阶和二阶导数的复杂公式来确定其高斯（如果dim M= 2）或标量（如果dim M> 3）曲率K。我们把这个公式表示为
In [9] we precisely described the Gaussian curvature functions on a compact connected 2-manifold M and gave related information on scalar curvatures in the higher dimensional case. We refel the reader to the introductions of [6, 8, 9] for historical and bibliographical information. Here we would like to present a slightly different proof of the above results. While the basic outline of the proof is almost identical to that in [9], it differs in that it takes the more direct approach of Fischer-Marsden [4] and does not use conformal deformations at all. This direct method is both technically and conceptually simpler and should be more accessible to differential geometers. On the other hand, there is a penalty in that one does not obtain the conformal information of our previous work [9], especially the deeper and more delicate information on pointwise conformal deformations [6, 8, 12]. Given a metric g on M, one can determine its Gaussian (if dim M= 2) or scalar (if dim M> 3) curvature K by a complicated formula involving the first and second derivatives of the metric. We abbreviate this formula as