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이멀젼은 로컬하게는 임베딩이 된다.

3173)이멀젼은 로컬하게는 임베딩이 된다.

정리¹

증명

$\phi$가 이멀젼이라고 가정했으므로, $d\phi{p}$가 일대일이다. 임베딩임을 보이려면, $\phi|_{V}$와 $(\phi|_{V})^{-1}$가 전단사여야하기 때문에 $\mathbb{R}^{m}$의 좌표를 $n$개까지만 $$ \begin{bmatrix} \dfrac{\partial y_{1}}{\partial x_{1}} & \dfrac{\partial y_{1}}{\partial x_{2}} & \dots & \dfrac{\partial y_{1}}{\partial x_{n}} \\[1em] \dfrac{\partial y_{2}}{\partial x_{1}} & \dfrac{\partial y_{2}}{\partial x_{2}} & \dots & \dfrac{\partial y_{2}}{\partial x_{n}} \\[1ex] \vdots & \vdots & \ddots & \vdots \\[1ex] \dfrac{\partial y_{m}}{\partial x_{1}} & \dfrac{\partial y_{m}}{\partial x_{2}} & \dots & \dfrac{\partial y_{m}}{\partial x_{n}} \end{bmatrix} $$

역함수 정리를 적용하기 위해, 다음의 매핑을 생각하자.

$$ \varphi : (U_{1} \times \mathbb{R}^{m-n=k})\subset \mathbb{R}^{m} \to \mathbb{R}^{m} $$

$$ \begin{align*} &\ \varphi ( x_{1}, \dots, x_{n}, t_{1}, \dots, t_{k} ) \\ =&\ \Big( y_{1}(x_{1},\dots,x_{n}), \dots, y_{n}(x_{1},\dots,x_{n}), y_{n+1}(x_{1},\dots,x_{n}) + t_{1}, \dots, y_{n+k}(x_{1},\dots,x_{n}) + t_{k} \Big) \end{align*} $$

$$ \begin{align*} \det( d \varphi _{q}) =&\ \dfrac{\partial (y_{1}, \dots, y_{n}, y_{n+1}, \dots, y_{n+k})}{\partial (x_{1}, \dots, x_{n}, t_{1}, \dots, t_{k})} \\ =&\ \begin{bmatrix} \dfrac{\partial y_{1}}{\partial x_{1}} & \dfrac{\partial y_{1}}{\partial x_{2}} & \dots & \dfrac{\partial y_{1}}{\partial x_{n}} \\[1em] \dfrac{\partial y_{2}}{\partial x_{1}} & \dfrac{\partial y_{2}}{\partial x_{2}} & \dots & \dfrac{\partial y_{2}}{\partial x_{n}} \\[1ex] \vdots & \vdots & \ddots & \vdots \\[1ex] \dfrac{\partial y_{m}}{\partial x_{1}} & \dfrac{\partial y_{m}}{\partial x_{2}} & \dots & \dfrac{\partial y_{m}}{\partial x_{n}} \end{bmatrix} \\ =&\ \end{align*} $$