Tag: 子流形

假设$(M^m,g)\hookrightarrow(\mathbb R^n,\bar g)$是嵌入到欧氏空间中的一个子流形. 又设其局部参数表示为$X(u^1,\ldots,u^m)=\bigl(x^1(u^1,\ldots,u^m),\ldots,x^n(u^1,\ldots,u^m)\bigr)$. 有以下两件事情: 首先, $X$在局部坐标下可视为$M\to\mathbb R^n$的一个映射, 而$(M,g)$的诱导度量(因为是嵌入)就是 $$ g=X^*(\bar g) $$ 即 \begin{align*} g_{\alpha\beta}&=g\left(\frac{\partial}{\partial u^\alpha},\frac{\partial}{\partial u^\beta}\right) =X^*(\bar g)\left(\frac{\partial}{\partial u^\alpha},\frac{\partial}{\partial u^\beta}\right) =\bar g\left(\frac{\partial X^i}{\partial u^\alpha}\frac{\partial}{\partial x^i},\frac{\partial X^j}{\partial u^\beta}\frac{\partial }{\partial x^j}\right)\\ &=\frac{\partial X^i}{\partial u^\alpha}\frac{\partial X^j}{\partial u^\beta}\delta_{ij} =\frac{\partial X^i}{\partial u^\alpha}\frac{\partial X^i}{\partial u^\beta}. \end{align*} 这即是我们常说的第一基本型.