Properties of $C^1$-smooth mappings with one-dimensional gradient range

We find necessary and sufficient conditions for a curve in $\mathbb R^{m\times n}$ to be the gradient range of a $C^1$-smooth function $v\colon\Omega\subset\mathbb R^n\to\mathbb R^m$. We show that this curve has tangents in a weak sense; these tangents are rank 1 matrices and their directions constitute a function of bounded variation. We prove also that in this case $v$ satisfies an analog of Sard's theorem, while the level sets of the gradient mapping $\nabla v\colon\Omega\to\mathbb R^{m\times n}$ are hyperplanes.