Riemannian metric on a level set of a smooth function on a manifold Also asked here: https://math.stackexchange.com/questions/1725491/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!
 A: First you need your level set $L \subset M$ to be a smooth manifold. For example you can require $G$ to be a submersion or that all points of $L$ are regular points for the map $G$.
Then, for all $q \in L$, $T_q L = \mathrm{ker} d_q G \subset T_q M$, where $d_q G : T_q M \to T_0\mathbb{R} = \mathbb{R}$.
The Riemannian structure $h$ on $L$ is just the restriction to the submanifold:
$$h_q(v,w) = g_q(v,w), \qquad \forall q \in L, \, \forall v,w \in T_q L \subset T_q M.$$
Remark: you cannot express $h$ using $G$ in a direct way (or at least, there is no explicit formula which gives $h$ in terms of $g$ and $G$). Indeed the metric $h$ is intrinsically defined by the restriction of $g$ to the submanifold $L$, and there are infinitely many ways to describe a fixed submanifold $L$ as the level set of a regular function.
As Ben suggested, choose a neighborhood $U$ of $q \in L$ and submanifold coordinates $(x,y) \in \mathbb{R}^{n-d} \times \mathbb{R}^d$ on $U$ such that $G(x,y) = y$. In this way:
$$ L \cap U \simeq \{ (x,0) \mid x \in \mathbb{R}^{n-d}\}. $$
Then the symmetric matrix representing $h$ on $L$ is just the $n-d\times n-d$ upper left block representing $g$, restricted to $L\cap U$:
$$h_x = \sum_{i,j=1}^{n-d} g_{ij}(x,0) \,dx^i \otimes dx^j. $$
