Let $M^n$ be a smooth Riemannian $n$-manifold and $f\in C^{\infty}(M;\mathbb{R})$. Let $p\in M$ be such that $\operatorname{grad}_p f\neq 0$ and $f(p)=c$. I want to prove that $f^{-1}(c)$ defines a $(n-1)$-submanifold on a neighborhood of $p$. For this purpose, I'm trying to prove that $f$ is submersive at $f^{-1}(c)$, i.e., that $D_pf:T_pM\rightarrow T_c\mathbb{R}$ is onto. However, I'm having problems trying to prove this. I've tried directly and via factoring $f$ through a chart, but I got stucked in both tries.
Could you guys lend me hand? I'd really appreciate it!
Thank you!