问题及解答

Prop. 设 $X,T$ 分别为流形 $M$ 上的向量场和 $(0,p)$-型张量, 则

\[
(L_X T)(Y_1,\ldots, Y_p)=D_X\bigl(T(Y_1,\ldots, Y_p)\bigr)-\sum_{i=1}^{p}T(Y_1,\ldots,L_X Y_i,\ldots, Y_p).
\]

写成局部坐标的形式, 以 (0,2)-型张量为例

\[
(L_X T)^{jk}=X^i\frac{\partial T^{jk}}{\partial x^i}-T^{ik}\frac{\partial X^j}{\partial x^i}-T^{ji}\frac{\partial X^k}{\partial x^i}.
\]

先证明 $p=1$ 的情形. (一般情形是类似的, 但需要更多的记号.) 利用

\[
Y|_{F^t}=DF^t(Y)+tDF^t(L_X Y)+o(t),
\]

得到

\[
\begin{split}
\bigl((F^t)^* T\bigr)&=T(DF^t(Y))\\
&=T(Y|_{F^t}-tDF^t(L_X Y))+o(t)\\
&=T(Y)\circ F^t -tT(DF^t(L_X Y))+o(t)\\
&=T(Y)+tD_X(T(Y))-tT(DF^t(L_X Y))+o(t).
\end{split}
\]

于是

\[
\begin{split}
(L_X T)(Y)&=\lim_{t\rightarrow 0}\frac{\bigl((F^t)^* T\bigr)(Y)-T(Y)}{t}\\
&=\lim_{t\rightarrow 0}\Bigl(D_X(T(Y))-T(DF^t(L_X Y))\Bigr)\\
&=D_X(T(Y))-T(L_X Y).
\end{split}
\]

$p=2$ 时.

\[
\begin{split}
\bigl((F^t)^* T\bigr)(Y_1,Y_2)&=T(DF^t(Y_1),DF^t(Y_2))\\
&=T(Y_1,Y_2)+t(L_X T)(Y_1,Y_2)+o(t).
\end{split}
\]

由定义,

\[
(L_X T)(Y_1,Y_2):=\lim_{t\rightarrow 0}\frac{\bigl((F^t)^* T\bigr)(Y_1,Y_2)-T(Y_1,Y_2)}{t}.
\]

\[
\begin{split}
T(DF^t(Y_1),DF^t(Y_2))&=T\Bigl(Y_1|_{F^t}-tDF^t(L_X Y_1)+o(t),\,Y_2|_{F^t}-tDF^t(L_X Y_2)+o(t)\Bigr)\\
&\stackrel{write}{=}T_1(Y_1|_{F^t}-tDF^t(L_X Y_1)+o(t))\cdot T_2(Y_2|_{F^t}-tDF^t(L_X Y_2)+o(t))\\
&=\Bigl[T_1(Y_1)\circ F^t -tT_1(DF^t(L_X Y_1))+o(t)\Bigr]\cdot\Bigl[T_2(Y_2)\circ F^t -tT_2(DF^t(L_X Y_2))+o(t)\Bigr]\\
&=\bigl(T_1(Y_1)\cdot T_2(Y_2)\bigr)\circ F^t -tT_1(DF^t(L_X Y_1))T_2(Y_2)\circ F^t-t(T_1(Y_1)\circ F^t)\cdot T_2(DF^t(L_X Y_2))+o(t)\\
&=T(Y_1,Y_2)\circ F^t-tT_1(DF^t(L_X Y_1))T_2(Y_2)\circ F^t-t(T_1(Y_1)\circ F^t)\cdot T_2(DF^t(L_X Y_2))+o(t)\\
&=T(Y_1,Y_2)+tD_X(T(Y_1,Y_2))+o(t)-tT_1(DF^t)(L_X Y_1))\Bigl[T_2(Y_2)+tD_X(T_2(Y_2))+o(t)\Bigr]\\
&\qquad -t\Bigl[T_1(Y_1)+tD_X(T_1(Y_1))+o(t)\Bigr]\cdot T_2(DF^t(L_X Y_2))+o(t).
\end{split}
\]

故

\[
\begin{split}
(L_X T)(Y_1,Y_2)&=\lim_{t\rightarrow 0}\frac{1}{t}\Bigl[((F^t)^* T)(Y_1,Y_2)-T(Y_1,Y_2)\Bigr]\\
&=\lim_{t\rightarrow 0}\Bigl[D_X (T(Y_1,Y_2))-T_1(DF^t(L_X Y_1))T_2(Y_2)-T_1(Y_1)T_2(DF^t(L_X Y_2))\Bigr]\\
&=D_X (T(Y_1,Y_2))-T_1(L_X Y_1)T_2(Y_2)-T_1(Y_1)T_2(L_X Y_2)\\
&=D_X (T(Y_1,Y_2))-T(L_X Y_1,Y_2)-T(Y_1,L_X Y_2).
\end{split}
\]

Remark: $p > 2$ 的情形类似地证明.