# Continuous quasi-isometry between Riemannian manifolds

geometric-topologymetric-geometryriemannian-geometry

Let $$(S,d_{1})$$ and $$(S',d_{2})$$ be two proper geodesic metric spaces. If there exists a quasi-isometric embedding $$f\colon S\rightarrow S'$$, does there exist a $$\textbf{continuous}$$ quasi-isometric embedding $$g\colon S\rightarrow S'$$ as well?.

If the answer is affirmative, is $$g$$ canonically related to $$f$$?. Concretely, is there a functor relating quasi-isometric embeddings to continuous quasi-isometric embeddings?.

If the answer is negative (which I think is the case), is the statement true at least for complete Riemannian surfaces (assuming quasi-isometries in the geometric group theoretical setting)?.

Edit:

A $$\textbf{quasi-isometric embedding}$$ $$f\colon S\rightarrow S'$$ is a function for which there exist positive constants $$L,C$$ such that
$$L^{-1}d_{1}(x,y)-C\leq d_{2}\big(f(x),f(y)\big)\leq Ld_{1}(x,y)+C$$
for all $$x,y\in S$$.

For a counterexample, take $$(S,d_1)$$ to be the Euclidean plane and $$(S',d_2)$$ to be the union of the integer coordinate lines in the Euclidean plane: $$S' = (\mathbb R \times \mathbb Z) \cup (\mathbb Z \times \mathbb R)$$
For a counterexample with complete Riemannian surfaces, start with the same $$(S,d_1)$$. For $$(S',d_2)$$, start with the Euclidean plane $$S$$, consider each integer coordinate square $$[m-1,m] \times [n-1,n]$$, and then for each $$m,n$$ replace that square by its connected sum with a torus.