If I have a smooth complete intersection of $f_1,\ldots,f_k \in C^\infty(\mathbb{R}^n)$, presented as the vanishing locus $$ f_1 = 0 \text{ } \cdots \text{ } f_k = 0 $$ in $\mathbb{R}^n$, how can I find the pullback of the standard euclidean metric under the inclusion map?
Consider the 4-manifold $M_1$ given by the vanishing of $$ \sin^3(x_1) - x_1^2\cdot x_3 + x_2^2 + x_4^3 + x_5^4 + 1 $$
Consider the 3-manifold $M_2$ given by the vanishing locus of $$ \begin{align*} x_1^2 + x_2^2 - x_3^4 + x_4\cdot x_5 + 1 \\ x_2^2 - x_1\cdot x_3 + x_4^4 + x_4\cdot x_5 \end{align*} $$ This is indeed a manifold since the jacobian is given by the matrix $$ \begin{bmatrix} 2x_1 & 2x_2 & -4x_3^3 & -x_5 & -x_4 \\ -x_3 & 2x_2 & -x_1 & 4x_3^3 + x_5 & x_4 \end{bmatrix} $$ which drops rank only at $(0,0,0,0,0)$