16
$\begingroup$

Let $(M^n,g)$ be a smooth complete Riemannian manifold. Let $p\in M$ be a point. Recall that the cut locus of $p$ is the set of vectors $v$ in the tangent space $T_pM$ such that $\exp(t v)$ is a minimizing geodesic for any $t\in [0,1]$, but not for $t\in [0,1+\varepsilon )$ for any $\varepsilon >0$.

Question. Does the cut locus have Lebesgue measure 0 in $T_pM$? If yes, does it have Hausdorff dimension at most $n-1$?

If the above questions have negative answers, one may ask the same questions about the exponential image of the cut locus.

Improve this question
$\endgroup$
3
  • 9
    $\begingroup$ See Sakai's "Riemannian geometry", lemma III.4.4. Namely if $\tilde C_p$ is the cut locus in $T_pM$ , and $C_p$ its emage under $exp$, then both $\tilde C_p$, $C_p$ are null sets, $dim (\tilde C_p)= n-1$ or $\tilde C_p$ is empty, and the equality becomes an inequality after applying $exp$. Here I think $dim$ is the covering dimension (but I have not thought enough about this). $\endgroup$
    – Igor Belegradek
    Commented Nov 28, 2016 at 13:03
  • $\begingroup$ @IgorBelegradek: Thanks! Why not to write it as a final answer? $\endgroup$
    – asv
    Commented Nov 28, 2016 at 14:39
  • 4
    $\begingroup$ Because you asked about Hausdorff dimension and I do not know much about this. I know Itoh, Tanaka and et el. worked on this, see projecteuclid.org/euclid.tmj/1178224899 but I have no time now to trace the references and write an intellegent answer. Would you do that yourself? $\endgroup$
    – Igor Belegradek
    Commented Nov 28, 2016 at 14:46

1 Answer 1

Reset to default
21
$\begingroup$

The key fact is that the cut time $t_c : UM \to \mathbb{R}$, defined on the unit tangent bundle $UM$ of a complete, $n$-dimensional Riemannian manifold, is locally Lipschitz continuous around all $v \in UM$ such that $t_c(v) < +\infty$. Hence the tangential cut locus at $p \in M$, that is $$ \tilde{C}_p = \{t_c(v)v\mid v \in UM,\quad t_c(v) < +\infty\} \subset T_q M, $$ either is empty, or it has Hausdorff dimension exactly $n-1$ (being the graph of a locally Lipschitz function). The exponential map being smooth, it cannot increase the Hausdorff dimension, hence $\dim(C_p) = \dim(\exp_p(\tilde{C}_p)) \leq n-1$.

All of this is proved here by Itoh and Tanaka.

Improve this answer
$\endgroup$
2
  • 1
    $\begingroup$ Is the cut-locus ever a sub-manifold? $\endgroup$
    – ABIM
    Commented May 19, 2020 at 22:57
  • 1
    $\begingroup$ @ABIM Just found this question. Even though I've forgot a lot of the differential geometry I studied, I'd day yes, as here: math.stackexchange.com/questions/364745/cut-locus-of-mathbbcpn, you'll see a proof of the cutlocus of $CP^n$ is $CP^{n-1}.$ Similar case for $RP^n.$ So yes to your question(?) $\endgroup$
    – Learning math
    Commented Aug 5, 2023 at 16:12

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .