Where is this Definition of a Manifold from?

Question

Let $M\subset \mathbb{R}^N$. We say that $M$ is a smooth submanifold of codimension $d$ or dimension $N-d=n$, if for every $p\in M$ there is a neighbourhood $U$ of $p$ and smooth functions $f_1,...,f_d$ on $U$ which have the properties:

1. $\mathrm{d}f_1,\mathrm{d}f_2,...,\mathrm{d}f_d$ are linearly independent in $T^*\mathbb{R}^N$
2. $U\cap M = \{f_1=...=f_d=0\} $

I have never seen this definition of a manifold and would like to know if there are any books working with it. I think this definition is closely related to the regular value theorem which is, again dependend on the exact definition of a manifold, a follow up theorem.

Answer 1

If I remember well, there is a similar definition in Berger /Gostiaux book Differential Geometry: manifolds, curves and surfaces..

I have (unfortunately not at hand) a paper copy of the original French book.

The idea is similar to the way to define a linear subspace using linear forms.