Also asked here: https://mathoverflow.net/questions/235163/riemannian-metric-on-a-level-set-of-a-smooth-function-on-a-manifold
Let $(M,g)$ be a finite or infinite dimensional Riemannian manifold. Let $G:M\to \mathbb{R}^d$ be a smooth map. Here $d < dim(M)$ if $M$ is finite dimensional, otherwise there's no restriction on $d$. Let $L:=G^{-1} \{0\} $. Is there a way we can express the restricted Riemannian metric on $L$ by explicitly using $G$?
So, let $q\in L, v,w\in T_q{L}$. How can we define the restricted inner product $g_q(v,w)$ on the submanifold $L$ using $G$, as explicitly as possible?
Thank you!