Is the injectivity radius constant if the Riemannian manifold has constant curvature?

Question

Suppose $M$ is a 2-dimensional complete Riemannian manifold of constant curvature. Is it true that the injectivity radius $i(p)$ is constant for all $p\in M$?

In all the examples I know, this is indeed the case. But I could not find the above result somewhere written. Does someone know the answer?

Best regards

Answer 1

If the curvature is non-negative the answer turns out to be yes, but in the hyperbolic case you can have, e.g., a surface of rotation of constant negative curvature having a thin neck. The injectivity radius is small where the neck pinches, larger elsewhere. (Such a metric extends to a complete metric of constant curvature, though the extended surface doesn't immerse in Cartesian $3$-space.)