Regular value theorem: Difference between revisions

Revision as of 23:35, 16 January 2008

Statement

Let $M, N$ be differential manifolds and $p \in N$ be a regular value of a differentiable map $f : M \to N$ . Then $f^{- 1} (p)$ is a submanifold of $M$ .

A slightly stronger version of this result states the following: if there is an open neighbourhood $U$ of $p$ in $N$ such that the rank of the Jacobian is constant for all points in $f^{- 1} (U)$ , then $f^{- 1} (p)$ is a submanifold of $M$ .

@@ Line 1: / Line 1: @@
 ==Statement==
-Let <math>M, N</math> be [[differential manifolds]] and <math>p \in N</math> be a [[regular value]] of a [[differentiable map]] <math>f: M \to N</math>. Then <math>f^{-1}(p)</math> is a submanifold of <math>M</math>.
+Let <math>M, N</math> be [[differential manifolds]] and <math>p \in N</math> be a [[regular value]] of a [[differentiable map]] <math>f: M \to N</math>. Then <math>f^{-1}(p)</math> is a [[submanifold]] of <math>M</math>.
+A slightly stronger version of this result states the following: if there is an open  neighbourhood <math>U</math> of <math>p</math> in <math>N</math> such that the rank of the Jacobian is constant for all points in <math>f^{-1}(U)</math>, then <math>f^{-1}(p)</math> is a submanifold of <math>M</math>.