How can I understand the regular point or the regular level set from a geometry view?

Question

I am learning manifold, when I see the regular level set theorem, I feel the great power of it. Although this theorem can help us to decide whether a geometry object is a manifold or not, I can't understand it from a geometry view, in the other words , I can't see the regular level set.

How can I see the regular level set ? Or what does a regular level set generally look like?

Answer 1

it is (by applying the implicit function theorem) locally a smooth deformation of a set like $$\{(x_1, \ldots, x_n): x_{i} = 0, 1 < k\le i\le n\}$$ This is, of course, just the special case of the zero level of $f(x) = (x_{k}, \ldots, x_n)$