Local Normal Form
If ƒ: M → N is a submersion at p and ƒ(p) = q ∈ N then there exist an open neighborhood U of p in M, an open neighborhood V of q in N, and local coordinates (x1,…,xm) at p and (x1,…,xn) at q such that ƒ(U) = V and the map ƒ in these local coordinates is the standard projection
It follows that the full pre-image ƒ−1(q) in M of a regular value q ∈ N under a differentiable map ƒ: M → N is either empty or is a differentiable manifold of dimension dim M − dim N, possibly disconnected. This is the content of the regular value theorem (also known as the submersion theorem). In particular, the conclusion holds for all q ∈ N if the map ƒ is a submersion.
Read more about this topic: Submersion (mathematics)
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