Riemann接続がaffine接続の座標変換則をみたすこと

　この記事は、共立出版　藤原彰夫著　「情報幾何学の基礎～情報の内的構造を捉える新たな地平～」での定義に準ずることにする。添え字ミスがあったら教えてください。インターネットで検索してもなかなかこんな面倒なことを公開している記事が見つからなかったので書きました。ちなみに上記の教科書では「読者の演習問題」になっています。

Prove that Christoffel symbols transformation law via the metric tensor

これを参考にした。

各種ざっくり定義

　多様体Mの各座標近傍に、以下の座標変換則

$$
\Gamma_{ij}{}^{k}=\dfrac{\partial\xi^{a}}{\partial x^{i}}\dfrac{\partial\xi^{b}}{\partial x^{j}}\dfrac{\partial x^{k}}{\partial \xi^{c}}\Gamma_{ab}{}^{c}+\dfrac{\partial^{2}\xi^{c}}{\partial x^{i}\partial x^{j}}\dfrac{\partial x^{k}}{\partial \xi^{c}}
$$

をみたす$${C^{\infty}}$$級関数の組$${\left\{\Gamma_{ij}{}^{k}\right\}}$$を与えることを、Mにaffine接続を与えるという。

　Mの(0,2)型テンソル場$${g}$$に対して、$${\forall p \in M}$$で$${g_{p}}$$が$${T_{p}(M)}$$の内積(正定値対称双線型形式)であるとき$${g}$$をRiemann計量という。

　この$${g}$$に対して、$${\left\{\Gamma_{ij}{}^{l}\right\}}$$を

$$
\Gamma_{ij}{}^{l}g_{lk}=\dfrac{1}{2}\left(\partial_{i}g_{jk}+\partial_{j}g_{ki}-\partial_{k}g_{ij}\right)
$$

と定める。$${g_{ij}}$$の逆行列$${g^{ij}}$$を用いて、

$$
\Gamma_{ij}{}^{l}=\dfrac{1}{2}g^{kl}\left(\partial_{i}g_{jk}+\partial_{j}g_{ki}-\partial_{k}g_{ij}\right)
$$

となる。これをRiemann接続またはLevi-Civita接続という。これが座標変換則をみたすことを示す。

準備

　Mの二つの座標近傍$${(U;x^{i})}$$、$${(V;\xi^{a})}$$を$${U\cap V \neq \varnothing}$$となるようにとる。$${x}$$に対して添え字$${i,j,k,l,m,n,\dotsc}$$を、$${\xi}$$に対して$${a,b,c,d,e,f,\dotsc}$$を使う。

$$
g_{ij} = g\left(\dfrac{\partial}{\partial x^{i}},\dfrac{\partial}{\partial x^{j}}\right) = g\left(\dfrac{\partial \xi^{a}}{\partial x^{i}}\dfrac{\partial}{\partial \xi^{a}},\dfrac{\partial \xi^{b}}{\partial x^{j}}\dfrac{\partial}{\partial \xi^{b}}\right)
$$

$$
= \dfrac{\partial \xi^{a}}{\partial x^{i}}\dfrac{\partial \xi^{b}}{\partial x^{j}}g\left(\dfrac{\partial}{\partial \xi^{a}},\dfrac{\partial}{\partial \xi^{b}}\right) = \dfrac{\partial \xi^{a}}{\partial x^{i}}\dfrac{\partial \xi^{b}}{\partial x^{j}}g_{ab}
$$

より$${g_{ij} =\dfrac{\partial \xi^{a}}{\partial x^{i}}\dfrac{\partial \xi^{b}}{\partial x^{j}}g_{ab}}$$を得る。またこの逆行列は、$${g^{ij} =\dfrac{\partial x^{i}}{\partial \xi^{a}}\dfrac{\partial x^{j}}{\partial \xi^{b}}g^{ab}}$$である。実際

$$
g_{ij}g^{jk}=\dfrac{\partial \xi^{a}}{\partial x^{i}}\dfrac{\partial \xi^{b}}{\partial x^{j}}g_{ab}\dfrac{\partial x^{j}}{\partial \xi^{c}}\dfrac{\partial x^{k}}{\partial \xi^{d}}g^{cd}
= \dfrac{\partial \xi^{a}}{\partial x^{i}}\delta_{c}{}^{b}\dfrac{\partial x^{k}}{\partial \xi^{d}}g_{ab}g^{cd}
=\dfrac{\partial \xi^{a}}{\partial x^{i}}\dfrac{\partial x^{k}}{\partial \xi^{d}}g_{ab}g^{bd}
$$

$$
=\dfrac{\partial \xi^{a}}{\partial x^{i}}\dfrac{\partial x^{k}}{\partial \xi^{d}}\delta_{a}{}^{d}=
\dfrac{\partial \xi^{d}}{\partial x^{i}}\dfrac{\partial x^{k}}{\partial \xi^{d}}
=\delta_{i}{}^{k}
$$

となる。

$$
\partial_{i}g_{jk} = \dfrac{\partial}{\partial x^{i}}g_{jk} = \dfrac{\partial \xi^{a}}{\partial x^{i}}\dfrac{\partial}{\partial \xi^{a}}g_{jk} = \dfrac{\partial \xi^{a}}{\partial x^{i}}\dfrac{\partial}{\partial \xi^{a}}\left( \dfrac{\partial \xi^{b}}{\partial x^{j}}\dfrac{\partial \xi^{c}}{\partial x^{k}}g_{bc} \right)
$$

$$
=\dfrac{\partial \xi^{a}}{\partial x^{i}}\left( \dfrac{\partial x^{l}}{\partial \xi^{a}}\dfrac{\partial^{2} \xi^{b}}{\partial x^{l}\partial x^{j}}\dfrac{\partial \xi^{c}}{\partial x^{k}}g_{bc}+
\dfrac{\partial \xi^{b}}{\partial x^{j}}\dfrac{\partial x^{l}}{\partial \xi^{a}}\dfrac{\partial^{2} \xi^{c}}{\partial x^{l}\partial x^{k}}g_{bc}+
\dfrac{\partial \xi^{b}}{\partial x^{j}}\dfrac{\partial \xi^{c}}{\partial x^{k}}\partial_{a}g_{bc} \right)
$$

$$
=\delta_{i}{}^{l}\dfrac{\partial^{2} \xi^{b}}{\partial x^{l}\partial x^{j}}\dfrac{\partial \xi^{c}}{\partial x^{k}}g_{bc}+
\dfrac{\partial \xi^{b}}{\partial x^{j}}\delta_{i}{}^{l}\dfrac{\partial^{2} \xi^{c}}{\partial x^{l}\partial x^{k}}g_{bc}+
\dfrac{\partial \xi^{a}}{\partial x^{i}}\dfrac{\partial \xi^{b}}{\partial x^{j}}\dfrac{\partial \xi^{c}}{\partial x^{k}}\partial_{a}g_{bc}
$$

$$
=\dfrac{\partial^{2} \xi^{b}}{\partial x^{i}\partial x^{j}}\dfrac{\partial \xi^{c}}{\partial x^{k}}g_{bc}+
\dfrac{\partial^{2} \xi^{c}}{\partial x^{i}\partial x^{k}}\dfrac{\partial \xi^{b}}{\partial x^{j}}g_{bc}+
\dfrac{\partial \xi^{a}}{\partial x^{i}}\dfrac{\partial \xi^{b}}{\partial x^{j}}\dfrac{\partial \xi^{c}}{\partial x^{k}}\partial_{a}g_{bc}
$$

$$
=\dfrac{\partial^{2} \xi^{b}}{\partial x^{i}\partial x^{j}}\dfrac{\partial \xi^{c}}{\partial x^{k}}g_{bc}+
\dfrac{\partial^{2} \xi^{b}}{\partial x^{i}\partial x^{k}}\dfrac{\partial \xi^{c}}{\partial x^{j}}g_{cb}+
\dfrac{\partial \xi^{a}}{\partial x^{i}}\dfrac{\partial \xi^{b}}{\partial x^{j}}\dfrac{\partial \xi^{c}}{\partial x^{k}}\partial_{a}g_{bc}
$$

$$
=\dfrac{\partial^{2} \xi^{b}}{\partial x^{i}\partial x^{j}}\dfrac{\partial \xi^{c}}{\partial x^{k}}g_{bc}+
\dfrac{\partial^{2} \xi^{b}}{\partial x^{i}\partial x^{k}}\dfrac{\partial \xi^{c}}{\partial x^{j}}g_{bc}+
\dfrac{\partial \xi^{a}}{\partial x^{i}}\dfrac{\partial \xi^{b}}{\partial x^{j}}\dfrac{\partial \xi^{c}}{\partial x^{k}}\partial_{a}g_{bc}
$$

$$
=\left(\dfrac{\partial^{2} \xi^{b}}{\partial x^{i}\partial x^{j}}\dfrac{\partial \xi^{c}}{\partial x^{k}}+
\dfrac{\partial^{2} \xi^{b}}{\partial x^{i}\partial x^{k}}\dfrac{\partial \xi^{c}}{\partial x^{j}}\right)g_{bc}+
\dfrac{\partial \xi^{a}}{\partial x^{i}}\dfrac{\partial \xi^{b}}{\partial x^{j}}\dfrac{\partial \xi^{c}}{\partial x^{k}}\partial_{a}g_{bc}
$$

より$${\partial_{i}g_{jk} = \left(\dfrac{\partial^{2} \xi^{b}}{\partial x^{i}\partial x^{j}}\dfrac{\partial \xi^{c}}{\partial x^{k}}+\dfrac{\partial^{2} \xi^{b}}{\partial x^{i}\partial x^{k}}\dfrac{\partial \xi^{c}}{\partial x^{j}}\right)g_{bc}+\dfrac{\partial \xi^{a}}{\partial x^{i}}\dfrac{\partial \xi^{b}}{\partial x^{j}}\dfrac{\partial \xi^{c}}{\partial x^{k}}\partial_{a}g_{bc}}$$を得る。途中で添え字の交換$${b \leftrightarrow c}$$と$${g_{bc}=g_{cb}}$$を用いた。$${(i,j,k)}$$に対して、$${(a,b,c)}$$の順になるように添え字をつけてやると見通しが良い。個人的に。

本番

まずRiemann接続の定義の$${()}$$内を作る。

$$
\partial_{i}g_{jk}+\partial_{j}g_{ki}+\partial_{k}g_{ij}
$$

$$
= \dfrac{\partial \xi^{a}}{\partial x^{i}}\dfrac{\partial \xi^{b}}{\partial x^{j}}\dfrac{\partial \xi^{c}}{\partial x^{k}}\partial_{a}g_{bc}+
\left(\dfrac{\partial^{2} \xi^{b}}{\partial x^{i}\partial x^{j}}\dfrac{\partial \xi^{c}}{\partial x^{k}}+\dfrac{\partial^{2} \xi^{b}}{\partial x^{i}\partial x^{k}}\dfrac{\partial \xi^{c}}{\partial x^{j}}\right)g_{bc}
$$

$$
+ \dfrac{\partial \xi^{b}}{\partial x^{j}}\dfrac{\partial \xi^{c}}{\partial x^{k}}\dfrac{\partial \xi^{a}}{\partial x^{i}}\partial_{b}g_{ca}+
\left(\dfrac{\partial^{2} \xi^{c}}{\partial x^{j}\partial x^{k}}\dfrac{\partial \xi^{a}}{\partial x^{i}}+\dfrac{\partial^{2} \xi^{c}}{\partial x^{j}\partial x^{i}}\dfrac{\partial \xi^{a}}{\partial x^{k}}\right)g_{ca}
$$

$$
- \dfrac{\partial \xi^{c}}{\partial x^{k}}\dfrac{\partial \xi^{a}}{\partial x^{i}}\dfrac{\partial \xi^{b}}{\partial x^{j}}\partial_{c}g_{ab}-
\left(\dfrac{\partial^{2} \xi^{a}}{\partial x^{k}\partial x^{i}}\dfrac{\partial \xi^{b}}{\partial x^{j}}+\dfrac{\partial^{2} \xi^{a}}{\partial x^{k}\partial x^{j}}\dfrac{\partial \xi^{b}}{\partial x^{i}}\right)g_{ab}
$$

$$
= \dfrac{\partial \xi^{c}}{\partial x^{k}}\dfrac{\partial \xi^{a}}{\partial x^{i}}\dfrac{\partial \xi^{b}}{\partial x^{j}}\left(\partial_{a}g_{bc}+\partial_{b}g_{ca}-\partial_{c}g_{ab}\right)
$$

$$
+\left( \dfrac{\partial^{2} \xi^{c}}{\partial x^{i}\partial x^{j}}\dfrac{\partial \xi^{a}}{\partial x^{k}}+\dfrac{\partial^{2} \xi^{c}}{\partial x^{i}\partial x^{k}}\dfrac{\partial \xi^{a}}{\partial x^{j}}+\dfrac{\partial^{2} \xi^{c}}{\partial x^{j}\partial x^{k}}\dfrac{\partial \xi^{a}}{\partial x^{i}}+\dfrac{\partial^{2} \xi^{c}}{\partial x^{j}\partial x^{i}}\dfrac{\partial \xi^{a}}{\partial x^{k}}-\dfrac{\partial^{2} \xi^{c}}{\partial x^{k}\partial x^{i}}\dfrac{\partial \xi^{a}}{\partial x^{j}}-\dfrac{\partial^{2} \xi^{c}}{\partial x^{k}\partial x^{j}}\dfrac{\partial \xi^{a}}{\partial x^{i}}\right)g_{ca}
$$

$$
=\dfrac{\partial \xi^{a}}{\partial x^{i}}\dfrac{\partial \xi^{b}}{\partial x^{j}}\dfrac{\partial \xi^{c}}{\partial x^{k}}\left(\partial_{a}g_{bc}+\partial_{b}g_{ca}-\partial_{c}g_{ab}\right)+2\dfrac{\partial^{2} \xi^{c}}{\partial x^{i}\partial x^{j}}\dfrac{\partial \xi^{a}}{\partial x^{k}}g_{ca}
$$

途中で添え字を変更して3項すべて$${g_{ca}}$$にした。$${a,b,c}$$に対して総和をとるので、各項ごとでシグマすると思えばOK。次で最後。

$$
\Gamma_{ij}{}^{l}=\dfrac{1}{2}g^{kl}\left(\partial_{i}g_{jk}+\partial_{j}g_{ki}-\partial_{k}g_{ij}\right)
$$

$$
=\dfrac{1}{2}g^{de}\dfrac{\partial x^{k}}{\partial \xi^{d}}\dfrac{\partial x^{l}}{\partial \xi^{e}}\left\{\dfrac{\partial \xi^{a}}{\partial x^{i}}\dfrac{\partial \xi^{b}}{\partial x^{j}}\dfrac{\partial \xi^{c}}{\partial x^{k}}\left(\partial_{a}g_{bc}+\partial_{b}g_{ca}-\partial_{c}g_{ab}\right)+2\dfrac{\partial^{2} \xi^{c}}{\partial x^{i}\partial x^{j}}\dfrac{\partial \xi^{a}}{\partial x^{k}}g_{ca}\right\}
$$

$$
=\dfrac{1}{2}g^{de}\dfrac{\partial x^{l}}{\partial \xi^{e}}\dfrac{\partial \xi^{a}}{\partial x^{i}}\dfrac{\partial \xi^{b}}{\partial x^{j}}\delta_{d}{}^{c}\left(\partial_{a}g_{bc}+\partial_{b}g_{ca}-\partial_{c}g_{ab}\right)+
g^{de}\dfrac{\partial x^{l}}{\partial \xi^{e}}\dfrac{\partial^{2} \xi^{c}}{\partial x^{i}\partial x^{j}}\delta_{d}{}^{a}g_{ca}
$$

$$
=\dfrac{\partial x^{a}}{\partial \xi^{i}}\dfrac{\partial \xi^{b}}{\partial x^{j}}\dfrac{\partial \xi^{l}}{\partial x^{e}}\dfrac{1}{2}g^{ce}\left(\partial_{a}g_{bc}+\partial_{b}g_{ca}-\partial_{c}g_{ab}\right)+
g^{ae}\dfrac{\partial x^{l}}{\partial \xi^{e}}\dfrac{\partial^{2} \xi^{c}}{\partial x^{i}\partial x^{j}}g_{ca}
$$

$$
=\dfrac{\partial x^{a}}{\partial \xi^{i}}\dfrac{\partial \xi^{b}}{\partial x^{j}}\dfrac{\partial \xi^{l}}{\partial x^{e}}\Gamma_{ab}{}^{e}+
\dfrac{\partial^{2} \xi^{c}}{\partial x^{i}\partial x^{j}}\dfrac{\partial x^{l}}{\partial \xi^{e}}\delta_{c}{}^{e}
$$

$$
=\dfrac{\partial x^{a}}{\partial \xi^{i}}\dfrac{\partial \xi^{b}}{\partial x^{j}}\dfrac{\partial \xi^{l}}{\partial x^{c}}\Gamma_{ab}{}^{c}+
\dfrac{\partial^{2} \xi^{c}}{\partial x^{i}\partial x^{j}}\dfrac{\partial x^{l}}{\partial \xi^{c}}
$$

これは座標変換則の式である。