I am trying to understand an example from Thirring's Classical Mathematical Physics, 2nd ed., p. 14.
I want to understand how the condition on $M$ satisfies the condition for the implicit function theorem.
Let $f \colon \mathbb{R}^{(n-m)+m} \rightarrow \mathbb{R}^m$ is $C^1$. Then, according to Wikipedia, what is necessary is that $Y$ be invertible, where $(df)(\mathbf{a, b}) = [X | Y]$. But if I understand correctly, the condition on $M$ only guarantees that for every row in $Y$, there exists a non-zero entry. It may be the case that $Y$ be singular.
Is it necessary that $m<n$ for $f$ to make sense? Assuming that the conditions for the implicit function theorem are satisfied, how did we get an $n-1$ dimensional manifold? Shouldn't we get an $n-m$ dimensional manifold?