I'm trying to solve the following exercise (Algebraic Geometry over the Complex Number, Donu Arapura, Exercise 2.2.16):
Let $f_1, \dots, f_r$ be $C^{\infty}$ functions on $\mathbb{R}^n$, and let $X$ be the set of common zeros of these functions. Suppose that the rank of the Jacobian $\left( \frac{\partial f_i}{\partial x_j} \right)$ is $n - m$ at every point of $X$. Then, show that $X$ is an $m$-dimensional submanifold using the implicit function theorem (see [109, p. 41]). In particular, show that the sphere $x_1^2 + \cdots + x_n^2 = 1$ is a closed $(n-1)$-dimensional submanifold of $\mathbb{R}^n$.
The definition of submanifold presented in the text is the following:
Given an $n$-dimensional $C^{\infty}$ manifold $X$, a closed subset $Y \subset X$ is called a closed $m$-dimensional submanifold if, for any point $x \in Y$, there exists a neighborhood $U$ of $x$ in $X$ and a diffeomorphism to a ball $B \subset \mathbb{R}^n$ containing $0$ such that $Y \cap U$ maps to the intersection of $B$ with an $m$-dimensional linear subspace.
I’m not sure how to approach the first part of this exercise. I don’t see how the rank of the Jacobian matrix connects directly to the implicit function theorem. I know that the constant rank theorem is a consequence of the inverse function theorem (and likewise for the implicit function theorem), and it seems well-suited for this application. However, I'm not very familiar with the constant rank theorem, so I would prefer a solution that uses the implicit function theorem directly.
