The continuity of Injectivity radius - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T12:37:52Z http://mathoverflow.net/feeds/question/53381 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/53381/the-continuity-of-injectivity-radius The continuity of Injectivity radius Chih-Wei Chen 2011-01-26T18:25:03Z 2011-01-26T19:52:22Z <p>Dear all,</p> <p>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.</p> <p>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.</p> <p>My question is, <strong>can we prove the continuity of Inj on a non-compact manifold under some conditions?</strong> </p> <p>(I think that the weakest condition is to assume the finiteness of Inj.)</p> <p>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!</p> http://mathoverflow.net/questions/53381/the-continuity-of-injectivity-radius/53392#53392 Answer by Anton Petrunin for The continuity of Injectivity radius Anton Petrunin 2011-01-26T19:40:06Z 2011-01-26T19:52:22Z <p>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.)</p> <p>If $R&lt;\mathrm{InjRad}_p$ then one can construct a smooth metric on a sphere with an isometric copy of $B_R(p)$ inside.</p> <p>If there is a sequence of points $x_n\to x$ such that $$\lim\ \mathrm{InjRad}_{x_n}&lt; \mathrm{InjRad}_x,$$ apply above consruction for $R$ slightly smaller than $\mathrm{InjRad}_x$. You get a compact manifold with non-continuous InjRad.</p> <p>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.</p>