Consider a pseudo-Riemannian manifold $(M,g)$ with $\dim M=n$.
Let $\Sigma$ be an embedded submanifold with $\dim\Sigma=k$ and inclusion map $\iota:\Sigma\rightarrow M$. We then define the induced metric tensor as $\gamma=\iota^\ast g$ i.e. the pullback of the metric tensor of $M$. Given local coordinates $(x^1,...,x^n)$ for $M$ and $(y^1, ...,y^k)$ for $\Sigma$, $$\gamma_{ij}=(\iota^\ast g)_{ij}=\frac{\partial x^\mu}{\partial y^i}\frac{\partial x^\nu}{\partial y^j}g_{\mu\nu} \qquad i,j\in \{1,...n-1\}\quad \mu,\nu\in \{1,...n\}.\tag{1}$$ Where $x^\mu(y)$ parametrizes the submanifold.
Now, I want to relate $\det g$, the determinant of a $n\times n$ matrix, and $\det \gamma$, the determinant of a $k\times k$ matrix but looking at $(1)$ I don't know how to proceed. In fact, written in matrix form, that is a product of non square matrices. Is there some other - maybe smarter - way to express $\det{g}$ in terms of $\det\gamma$? The only thing I can expect is that the relation is dependent on our choice of coordinates as a consequence of $(1)$
It would be ok if someone had an idea just for the $k=n-1$ case (i.e. for hypersurfaces case).