**Setup/Motivation:**
Let $(M,g)$ and $(N,\rho)$ be complete Riemannian manifolds of respective dimensions $m$ and $n$ and suppose that $m\leq n$. Let $\operatorname{bi-C}^{\infty}(M,N)$ denote the class of bi-Lipschitz smooth maps from $M$ to $N$. When is $\operatorname{bi-C}^{\infty}(M,N)$ dense in $\operatorname{C}^{\infty}(M,N)$ in the *compact-open topology*?


**Update:** Ben's question affirms a simple "no" for the non-compact case. But (since I ask for when) if both $M$ and $N$ are compact then can we hope to obtain a positive answer?



Density of smooth bi-Lipschitz maps in smooth maps
Suppose that neither $M$ nor $N$ are compact and have dimensions both 2 or more.
Take an unbounded real valued function $f$ on $M$ and an unbounded geodesic $\gamma$ on $N$. Map $M$ to $N$ by taking each point $m \in M$ to the point $\gamma(t)$ where $t=f(m)$. This map is smooth but not approximable by any bi-Lipschitz map, since all $C^0$ nearby maps are not onto.


For $M$ and $N$ compact and diffeomorphic, clearly we can't approximate a constant map by a biLipschitz smooth map, i.e. a smooth map, in the compact open topology, as forcing the image of the map to lie in a tiny open set near a point of $N$ forces the map to be trivial on the fundamental class in homology, so not homotopic to a diffeomorphism.


