The continuity of Injectivity radius

Question

$\DeclareMathOperator{\InjRad}{InjRad}$ Dear all,

when reading a book of M. Berger, I learned that the injectivity radius $\InjRad(x)$ on a compact Riemannian manifold depends continuously on the point $x$.

When the manifold is complete and non-compact, $\InjRad$ may not be continuous. For example, $\InjRad(x)$ 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 $\InjRad$ on a non-compact manifold under some conditions?

(I think that the weakest condition is to assume the finiteness of $\InjRad$.)

PS. I must admit that I don't know how to prove the continuity of $\InjRad$ 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!

Answer 1

$\DeclareMathOperator{\InjRad}{InjRad}$ The compactness is irrelevant, it is proved in 5.4 of "Riemannsche Geometrie im Grossen" by Detlef Gromoll, Wilhelm Klingenberg, and Wolfgang Meyer.

But let me show that if it is true for compact manifolds, then the same is true for complete ones.

If $R<\InjRad_p$, 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 $$\lim \InjRad_{x_n}< \InjRad_x,$$ apply the above construction for $R$ slightly smaller than $\InjRad_x$. You get a compact manifold with non-continuous InjRad.

If there is a sequence of points $x_n\to x$ such that $$\lim\ \InjRad_{x_n}> \InjRad_x$$ then apply the above construction for $p=x_n$ for large enough $n$ and $R>\InjRad_x$. That leads to a contradiction again.