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Let $(M,g)$ be a Riemannian manifold, and $p\in M$. As $R>0$ increases, the topology of the ball $B(p,R)$ changes, but the changes happen only at a Lebesgue measure zero set of $R$. For instance, it can only change near $R$ when $R$ is a critical value for the distance function $d(p,\cdot)$, in the sense of Grove-Shiohama, while work of Rifford implies that a.e. $R$ is a regular value for $d(p,\cdot)$.

Question 1: Is it true that for a.e. $R$, when $\epsilon$ is small, there's a diffeomorphism between the open balls ${B(p,R)}$ and ${B(p,R+\epsilon)}$ that's $C^\infty$-close to the identity? (In particular, I don't want derivatives to blow up as you near the boundaries of the balls.)

If the boundaries of these balls were smooth codim 1 submanifolds, you should be able to do this by an easy isotopy, but I don't know how to do it in general, or if there's some meaningful piecewise-smooth structure on the boundary of (almost every) Riemannian metric ball.

Also, if the answer is yes and the proof is too easy, how about:

Question 2: Is it true that for a.e. $R$, if $g'$ is another Riemannian metric that is $C^\infty$-close to $g$, there's a diffeomorphism between the open balls ${B_{g}(p,R)}$ and ${B_{g'}(p,R)}$ that's $C^\infty$-close to the identity?

I'll accept answers to either question, obviously.

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  • $\begingroup$ I don't have counterexamples but I think what you want in Q1 is too good to be true. The usual critical point theory for the distance function only yields homeomorphism. One can smooth it to a diffeomorphism by approximation techniques (of Greene-Shiohama, see their 1981 paper with "differential structure" in the title) but I see no reason why the derivatives would survive the approximation. $\endgroup$
    – Igor Belegradek
    Jun 25, 2015 at 6:14
  • $\begingroup$ Hmm... noted. Thanks for the pointer to the Greene-Shiohama paper, though. I hadn't seen that string of papers yet. $\endgroup$
    – biringer
    Jun 25, 2015 at 12:20
  • $\begingroup$ You may be interested in this work by Herzog and Wilhelm, which pushes the Grove-Shiohama concepts a bit further. arxiv.org/abs/1409.0132 $\endgroup$
    – John Harvey
    Jun 25, 2015 at 20:15
  • $\begingroup$ Thanks for the link! I hadn't seen that extension of the (topological) analogy with Morse theory, so even though it's not so relevant to the smooth questions above, it's good to know for future reference. $\endgroup$
    – biringer
    Jun 26, 2015 at 13:29
  • $\begingroup$ Also, @Igor, after skimming through that paper it seems that the biggest difference is that my Qs are specifically about sublevel sets of the distance function, whereas (essentially) they just perturb to some smooth function and use the new function to say things about the manifold. In some sense, I think of my Qs as asking whether for generic R, the singularities in the boundary of an R-ball match those in the boundary of an R+eps ball, for small eps. $\endgroup$
    – biringer
    Jun 26, 2015 at 13:33

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