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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!

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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.

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