when reading a book of M. Berger, I learned that the injectivity radius Inj(x) on a compact Riemannian manifold depends continuously on the point x.
When the manifold is complete and non-compact, Inj may not be continuous. For example, Inj(x) decreases to zero when x moves to the most curved point on a paraboloid. However, it could be infinity at that point.
My question is, can we prove the continuity of Inj on a non-compact manifold under some conditions?
(I think that the weakest condition is to assume the finiteness of Inj.)
ps. I must admit that I don't know how to prove the continuity of Inj even on a compact manifold. I think that the argument should involve the stability of ODEs (the geodesic equation and Jacobi equation). If one of you have a reference about this, could you please tell me? thanks a lot!