Regular value theorem

Statement

For a smooth map on smooth manifolds with and , let be a regular value. The level set is an embedded submanifold with codimension is equal to the dimension of the codomain (in other words ).

Proof

Let be a regular value. We know we can find charts and centered at and such that for open sets , , , and

This can be seen in the following diagram:

Thus, we can take the level set , and this image under (call it ) would all be in the kernel of . Thus, So This satisfies the definition of an submanifold. (Note that in the last equation, it should be just on an open set.)

Uses/ why we care

This is a very easy way to find submanifolds for some smooth manifold.