Past exam of the short options course of the University of Oxford / 2021 / 4 / c / ii / Solution

Suppose $D{F}_x$ were invertible at some $x\in \Omega$. The inverse function theorem supplies a neighborhood $V$ of $x$ for which $F(V)$ is a nonempty open subset of ${\mathbb{R}}^2$. But $F(V)\subset M$, and a one-dimensional embedded submanifold of ${\mathbb{R}}^2$ has empty interior. To see this last assertion, a slice chart for an embedded submanifold takes $M$ locally into a coordinate line; a coordinate line cannot contain an open plane neighborhood, and a chart preserves openness. This is a contradiction.

Thus $D{F}_x$ is singular everywhere. Part (c)(i) strengthens singularity to $D{F}_x=0$ everywhere. By (b) and connectedness of $\Omega$,The special form of the Jacobian matrix is essential: an arbitrary differentiable map can have nonconstant image in a curve with derivative of rank one.