Suppose I have a smooth, noncompact manifold $M$ with metrics $g\_i$ for $i = 1,2$. Suppose there exists a $C \geq 1$ such that
$$ C^{-1} d\_1(x,y) \leq d\_2(x,y) \leq C d\_1(x,y)$$ 
where $d\_i$ are the induced distance metric of $g\_i$. Then, if the injectivity radius with respect to $g\_1$ satisfies $\mathrm{inj}(M, g\_1) \geq \kappa > 0$, does it follow that $\mathrm{inj}(M, g\_2) \geq \kappa' > 0$ for some $\kappa'$? 


I think I've confused and misled some people by introducing the completely irrelevant $\kappa$ and $\kappa'$. Sorry about that. What I mean is that if $\mathrm{inj}(M,g\_1) > 0$ then does it follow that $\mathrm{inj}(M,g\_2) > 0$? 


Note also that $\mathrm{inj}(M,g)$ is the infimum of the injectivity radius with respect to $g$ at each point.



Preservation of injectivity radius
This is an expansion of Anton Petrunin's comment.


Let me describe how to perturb the standard metric of the plane so that the resulting metric is bi-Lipschitz to the original with Lipschitz constant arbitrarily close to 1 and has arbitrarily small injectivity radius.


Locate the plane in $\mathbb R^3$ as the $xy$-plane. Remove the unit disc centered at 0. Pick a point $(0,0,\delta)\in\mathbb R^3$ slightly above the origin and connect it by segments with the points of the boundary circle of the removed disc. This gives you a singular surface consisting of a planar part and a conical part. On the cone, there are arbitrarily short geodesic digons near the apex. Fix such a digon of small length $\rho>0$ and smoothen the surface in a so small neighborhood of the apex that it does not reach the digon. Also smoothen the surface near the circle. Now we have a smooth surface with injectivity radius $\le\rho$ because the geodesic digon is still there.


The projection of this surface to the plane is a bi-Lipschitz diffeomorphism with bi-Lipschitz constant $C=\sqrt{1+\delta^2}$. This map sends the surface's metric to a new Riemannian metric on the plane, and it is $C$-bi-Lipschitz to the original one.


To construct a metric with zero injectivity radius, just repeat this construction in a countable collection of disjoint discs $B\_i\subset\mathbb R^2$, $i=1,2,\dots$, using the parameter $\rho=\rho\_i\to 0$.


