Dear all,


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!



The continuity of Injectivity radius
The compactness is irrelevant; i.e if it is true for compact manifolds then the same is true for complete ones. (The same proof as in comact case works, but it is easier to do this way.)


If $R<\mathrm{InjRad}\_p$ then one can construct a smooth metric on a sphere with an isometric copy of $B\_R(p)$ inside.


If there is a sequence of points $x\_n\to x$ such that
$$\lim\ \mathrm{InjRad}\_{x\_n}< \mathrm{InjRad}\_x,$$
apply above consruction for $R$ slightly smaller than $\mathrm{InjRad}\_x$.
You get a compact manifold with non-continuous InjRad.


If there is a sequence of points $x\_n\to x$ such that 
$$\lim\ \mathrm{InjRad}\_{x\_n}> \mathrm{InjRad}\_x$$ then apply above construction for $p=x\_n$ for large enough $n$ and $R>\mathrm{InjRad}\_x$.
That leads to a contradiction again.


