凯勒度量[Kähler Metric] - 达人(其实～我是卧底，来自外星)

2005/04/15 | 凯勒度量[Kähler Metric]
类别(∑〖数学〗) | 评论(0) | 阅读(39) | 发表于 19:59: A Kähler metric is a Riemannian metric g on a complex manifold which gives M a Kähler structure, i.e., it is a Kähler manifold with a Kähler form. However, the term "Kähler metric" can also refer to the corresponding Hermitian metric , where is the Kähler form, defined by . Here, the operator J is the almost complex structure, a linear map on tangent vectors satisfying , induced by multiplication by i. In coordinates , the operator J satisfies and
The operator J depends on the complex structure, and on a Kähler manifold, it must preserve the Kähler metric. For a metric to be Kähler, one additional condition must also be satisfied, namely that it can be expressed in terms of the metric and the complex structure. Near any point p, there exists holomorphic coordinates such that the metric has the form

where denotes the vector space tensor product; that is, it vanishes up to order two at p. Hence, any geometric equation in involving only the first derivatives can be defined on a Kähler manifold. Note that a generic metric can be written to vanish up to order two, but not necessarily in holomorphic coordinates, using a Gaussian coordinate system.; 0