Suppose $f_1,\dots, f_d$ are a set of real-valued functions on a smooth manifold $M$. Let $N$ be the zero locus of the $f_i$. Suppose the $\textrm{d}f_i$ span a subspace of the cotangent space of $M$ of dimension $d'<d$. I'm trying to prove that this gives $N$ a natural manifold structure.
I've applied the inverse function theorem to prove that wlog the $f_1,\dots,f_d'$ are local coordinates around any given $p$ in $M$. I'd now like to restrict my diffeomorphism to $N$, but I'm worried that the extra $f_i$ for $i > d'$ will yield some subtleties. In particular my lecturer suggested that I needed to check that the other $f_i$ were constant on all curves through $p$. Why is this, and how does it help? I'm afraid I haven't quite got the intuition for this yet, so any comments would be gratefully appreciated!
What you need is the Constant Rank Theorem, which is an equivalent formulation of the Inverse Function Theorem.
If $f:M\to N$ is a smooth map having constant rank then, for any $p\in M,$ there exist coordinate maps $\phi$ around $p$ on $M,$ and $\psi$ around $f(p)$ on $N,$ such that $$\psi\circ f\circ\phi^{-1}:(u,v)\in U\times V\to (u,0)\in U\times W,$$ where $U,V,W$ are open neighborhood of $0$ in Euclidean spaces.