## Return to Answer

2 Edited in line with Tom Goodwillie's comment.

Elementary but still useful is the regular value theorem or the submersion theorem:

Let $f \colon M \to N$ be smooth and $n \in N$. If $T(f)_m \colon T_m M \to T_{f(m)} N$ is onto for all $m \in f^{-1}(n)$ then $f^{-1}(n) \subset M$ is a submanifold of dimension $\text{dim}(N) - \text{dim}(M)$ or empty. text{dim}(M)$. 1 [made Community Wiki] Elementary but still useful is the regular value theorem or the submersion theorem: Let$f \colon M \to N$be smooth and$n \in N$. If$T(f)_m \colon T_m M \to T_{f(m)} N $is onto for all$m \in f^{-1}(n)$then$f^{-1}(n) \subset M$is a submanifold of dimension$\text{dim}(N) - \text{dim}(M)\$ or empty.