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Let $M$ be a complete Riemannian manifold with a point $p  \in M$ and let $U \subset T_pM$ be an open disk containing $0_p$ in the tangent space to $p$. By $C_p$ we denote the image of the boundary of $U$ under the exponential map, i.e. $$ C_p := \exp_p(\partial U). $$ If you want, you can think of $C_p$ as the cut locus at $p$.

Now, $C_p$ is a null-set (a set of measure zero) because $\partial U$ is a null-set and $exp_p$ is a smooth map. Let $f:M \rightarrow M$ be any homeomorphism on $M$.

My question is now: Is $\ f(C_p)$ also a null-set?

It is clear that in general the image of a null-set under a homeomorphism does not have to be a null-set. However, examples for this always seem to be mappings between Cantor sets and fat Cantor sets, which are highly disconnected.

Thank you!

wabu wabu rabu
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No, not in general. For example, the image of a circle in a plane under a homeomorphism of athe plane can be a curve of positive area. Indeed, here you can see the construction of a simple curve of positive area. It can be made closed by adding an arc, and then, by the Schoenflies theorem, it is the image of a circle under a homeomorphism of $\mathbb R^2$.

You can apply the above to the cut locus as well: e.g., for a point on a torus, the cut locus is a circle.

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