Below is a paragraph from the appendix from Krantz's Several Complex Variables book. I have limited knowledge regarding manifolds and was hoping (very much) that someone would be willing to provide the theorems used in each step and cite the theorem (hopefully from a common textbook). Help is appreciated very much!
Let $\Omega \subseteq \mathbb{R}^N$ have $C^k$ boundary. Then $M = \partial \Omega$ is the zero set of a $C^k$ defining function $\phi$. Therefore the implicit function theorem makes it clear that $M$ is a $C^k$ manifold of dimension $N-1$.