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.

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    $\begingroup$ 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 Nov 28 '16 at 13:03
  • $\begingroup$ @IgorBelegradek: Thanks! Why not to write it as a final answer? $\endgroup$ – MKO Nov 28 '16 at 14:39
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    $\begingroup$ 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 Nov 28 '16 at 14:46

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.

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  • $\begingroup$ Is the cut-locus ever a sub-manifold? $\endgroup$ – AIM_BLB May 19 at 22:57

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