We know that in $\mathbb{R}^2$, an implicit curve can be described by $$F(x,y)=0\,,$$ where $F$ satisfies certain regularity conditions, and normal / tangential vectors can be derived explicitly. I have been looking for a generalisation of the implicit description to higher dimensions, and in particular in terms of differential forms.
The idea comes intuitively from the fact that for $(x^1, \ldots, x^n) \in \mathbb{R}^n$, an implicit hypersurface $\mathcal{M}$ can be described by $F(x^i) = 0$ with additional regularity condition, where $\dim \mathcal{M} = n-1$.
One is tempted to generalise the claim to a set of "regular" functions $F_1(x^i), \ldots, F_m(x^i)$, $m < n$, such that $$F_a(x^i) = 0\,, \qquad a = 1, \dots, m\,,$$ describes an $(n-m)$-dimensional submanifold. Two cases with $m=1$ have been described above. A curve, which is a $1$-dimensional submanifold, can therefore be described by $m=n-1$ such equations.
My guess is that this could be summarised by the description $\mathrm{d} f \wedge \mathrm{d} g = 0$, where $f$ and $g$ are scalar functions on a real differential manifold $\Sigma$, $\dim \Sigma = n$. The degeneracy condition contains exactly $(n-1)$ equations in terms of the components.
Does this implicit description of curves exists in the literature? In particular, I would like to know about the tangential / normal vectors of the implicit curve.