Does there exist "the implicit function (existence) theorem" in the differential manifold theory?
We have known that there exist inverse function theorem and implicit function theorem in the Euclid space.
And I have known that in the differential manifold theory, there exist the inverse function theorem. However, does there exist "the implicit function existence theorem" or its generalization in the differential manifold theory?
Thanks a lot.
On
Perhaps you are thinking of the Regular Value Theorem? Which says, given a smooth map between smooth manifolds $f : M \to N$ of dimensions $m \ge n$, and given $p \in N$, suppose that $p$ is a regular value, meaning that for every $x \in f^{-1}(p)$ the derivative $Df_x : T_x M \to T_{p} N$ is surjective. Then $f^{-1}(p)$ is a sub manifold of $M$ of dimension $m-n$.
Because the implicit function theorem is purely local, it carries over just fine to differentiable manifolds, and is widely used. I think it may be described in some detail in Milnor's Topology from the Differentiable Viewpoint, but I could be mistaken. It's almost certainly discussed in Guillemin and Pollack's Differential Topology book.