微分几何笔记讨论班的粗略总结Chp7-5

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2008.10.3 & 2008.10.8
7.6 Isometries and conformal transformations
7.6.1 Isometries(等距映射)
Isometry can be looked as a coordinates trasnformation with a unchanged distance.
7.6.2 Conformal transformations
A conformal transformation means that a metric is unchanged up to a scale. If the map is from point p to point p, so the map can be called Weyl rescaling.
Weyl tensor is a invariant under the conformal transformation.
7.7 Killing vector fields and conformal Killing vector fields
7.7.1 Killing vector fields
Killing Vector fields is an infinitesimal form of isometry.
A one-parameter group of transofrmations which generates the Killing vector field X(p.190).
The Killing equation can be written in Lie derivative form.
In the m-D Minkowski spacetime, according to the Killing equation, we have 10 independent solutions. That is m(m+1)/2 Killing vector fields, it's called maximally symmetric spaces. We can know from this concept, the number of Killing vector fields can greater than the dimension of the manifold. From S^2 with the standard metric, and combined with Killing equation, we have three generators, Lx,Ly,Lz, these are the generators of Lie algebra so(3).
S^n with the usual metric is a maximally symmetric space.
7.7.2 Conformal Killing vector fields
Conformal Killing vector fields (CKV) is an infinitesimal form of Killing vector fields.
The metric is unchanged up to a scale.
Both CKV and killing vector fields have two properties:
1. linearity; 2. closure.
So both of them form a Lie algebra of symmetric operations on the manifold M.
Summarization
The definition of Killing vector fields and Killing equation is the most important thing in the context. And all the Killing vector fields form a Lie algebra.
A general Killing vector can be written as the linear combination of the bases of the corresponding Lie algebra.
Highlights
Exercise 7.17中的公式（7.121）的证明，T同学给出了一个很让人惊讶的提示，T同学不愧是高手啊～该公式的证明应用下面这个关系：
\partial_{\mu}X_{\nu=}(g_{\nu\lambda}X^{\lambda})=
g_{\lambda\nu}\partial_{\mu}X^{\lambda}+\partial_{\mu}g_{\nu\lambda}X^{\lmbda}, 然后应用公式（7.30）。

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