Completion of a vector-valued function to form a diffeomorphism

[Mathieu Le Provost]

Hello, 

I am wondering if someone can help me on this question:

Let $\varphi_r \colon \mathbb{R}^n \to \mathbb{R}^r$ be a $\mathcal{C}^1(\mathbb{R}^n, \mathbb{R}^r)$ vector-valued function.

Do we know sufficient conditions on $\varphi_r$ for the existence of a function $\varphi_\perp \colon \mathbb{R}^n \to \mathbb{R}^{n -r}$ such that the augmented function $\varphi \colon \mathbb{R}^n \to \mathbb{R}^n, x \mapsto (\varphi_r(x), \varphi_\perp(x))$ is a diffeomorphism on $\mathbb{R}^n$?

Thank you for your help, 

Mathieu

[Nicolas Boumal]

I don't know. One observation would be this: if f : R^n - R is strongly convex, then the gradient of f is a diffeomorphism from R^n to R^n. So, if  \varphi_r corresponds to $r$ components of the gradient of a strongly convex function, then it can be completed in the way you described. But that's not really more helpful than saying "if \varphi_r is part of a diffeomorphism, then it can be completed to a diffeomorphism" ...