I have some basic confusion around the relationship between injectivity radius and evenly covered neighborhoods.
Suppose a Riemannian manifold $M$ has positive injectivity radius $c_1>0$. Let $\widetilde{M}$ be a covering space of $M$. Then does it follow that there exists a constant $c_2>0$ such that each point $x\in M$ has a neighborhood of radius $c_2$ that is evenly covered by $\widetilde{M}$?