微分流形Loring Tu 习题21.2解答

今天的作业，随手写到博客吧.

\(Proof.\)对于任意的\(p \in M\)，有p附近的坐标卡\((U,x^{1},\ldots,x^{n})\), 由引理\(21.4\),$$dx^{1}\wedge\ldots \wedge dx^{n}(X_{1,p},\ldots,X_{n,p})>0$$

设\(\beta=dr^{1}\wedge\ldots \wedge dr^{n}\),

\[\beta (\frac{\partial }{\partial r^{1} },\ldots,\frac{\partial }{\partial r^{n} })=dr^{1}\wedge\ldots \wedge dr^{n}(\frac{\partial }{\partial r^{1} },\ldots,\frac{\partial }{\partial r^{n} })=1 \]

\[\begin{array} \beta (X_{1,p},\ldots,X_{n,p})&=&dr^{1}\wedge\ldots \wedge dr^{n}(\varphi_{*} X_{1,p},\ldots,\varphi_{*}X_{n,p}) \\ &=&dx^{1}\wedge\ldots \wedge dx^{n}(X_{1,p},\ldots,X_{n,p})>0 \end{array} \]

故\((\varphi_{*} X_{1,p},\ldots,\varphi_{*}X_{n,p}) \sim (\dfrac{\partial }{\partial r^{1} },\ldots,\dfrac{\partial }{\partial r^{n} })\). 证毕.