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2 | Edited in line with Tom Goodwillie's comment. | ||
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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)$. |
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1 | [made Community Wiki] | ||
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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. |
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