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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 $N-1$.

wellfedgremlin
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    It is not true that the zero set of a C^k function is a manifold —some sort of non-degeneracy condition is needed. (maybe an important piece of context is missing; for example, the precise meaning of «defining function»...) – Mariano Suárez-Álvarez May 08 '14 at 01:45
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    In any case, you should probably pick a textbook on smooth manifolds (Lee's is quite amazing) and read through the first few chapters. – Mariano Suárez-Álvarez May 08 '14 at 01:48

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