6
$\begingroup$

I recently wondered what are the spaces whose morphisms are Lipschitz maps (by which I mean: "locally Lipschitz"). The answer seems pretty clear, and proceeds like the definition of manifolds:

1) If $X$ is a topological space, a Lipschitz chart is a homeomorphism from an open subset of $X$ to a metric space. 2) Two Lipschitz charts are compatible if the corresponding two transition functions are Lipschitz. 3) A Lipschitz atlas on $X$ is a set of compatible Lipschitz charts whose domains cover $X$. 4) A Lipschitz space is a topological space equipped with a maximal Lipschitz atlas.

So my question is: what are these spaces called? (and why not "Lipschitz spaces"?). I'll be grateful for any reference.

$\endgroup$
  • 1
    $\begingroup$ "objects whose morphisms" and "spaces whose morphisms" both sound weird... $\endgroup$ – YCor Dec 27 '16 at 12:12
  • 2
    $\begingroup$ Lipschitz manifolds is quite common. $\endgroup$ – Pietro Majer Dec 27 '16 at 13:43
  • 2
    $\begingroup$ Outside of pathological cases it will generally be possible to find a single metric on $X$ whose restriction to each chart is bi-Lipschitz equivalent to the metric on that chart. So unless you care about the pathological exceptions, you're just talking about metric spaces. "Lipschitz spaces" is not a good term because it is already used to refer to spaces of Lipschitz functions on a metric space, which are functional analytic objects. $\endgroup$ – Nik Weaver Dec 27 '16 at 16:12
  • $\begingroup$ @Nik You're right, and "non pathological" probably means "paracompact Hausdorff", here, but it seems unnatural to single out a particular metric. The space I'm actually interested in (a sort of "space of shapes") might actually be a Lipschitz Banach manifold, but this looks hard to prove, and I only need a well-defined notion of Lipschitz map into it. $\endgroup$ – Benoit Jubin Dec 27 '16 at 17:17
  • $\begingroup$ It sounds like what you want is something like "metric space, up to bi-Lipschitz modification"? Maybe uniform structure is what you want, then, although the natural maps are then uniform, not Lipschitz. $\endgroup$ – Nik Weaver Dec 27 '16 at 18:03
2
$\begingroup$

I randomly found an answer to my question two years later! The spaces I described in the question are called "locally metric spaces" in

J. Luukkainen, J. Väisälä, Elements of Lipschitz topology, Ann. Acad. Sci. Fennicae 3 (1977), 85--122.

They are introduced, together with "local metrics", in their Section 3.4. As guessed in the comments above, any paracompact Hausdorff locally metric space is indeed a metric space "up to Lipschitz equivalence" (their Theorem 3.5). They cite

J. H. C. Whitehead, Manifolds with transverse fields in Euclidean space, Ann. Math. 73 (1961), 154--212.

which introduces "local metrics" in Section 2.

Note: to reply to YCor's comment: indeed, I should have written "objects of the category whose morphisms" instead of "objects whose morphisms".

$\endgroup$
  • 1
    $\begingroup$ Link to paper: acadsci.fi/mathematica/Vol03/vol03pp085-122.pdf $\endgroup$ – YCor Jan 20 at 18:21
  • $\begingroup$ To be complete: a local metric on a topological space is defined as the data of an open covering $(U_i)$, a metric $d_i$ on each $U_i$, such that the identity map $(U_i\cap U_j,d_i)\to (U_i\cap U_j,d_j)$ is Lipschitz for each $i$. There is a natural equivalence of local metrics (local bilipschitz equivalence). A locally metric space is defined there as a topological space endowed with an equivalence class of local metrics. Of course it's important to emphasize "local" since many people are interested in global or even large-scale aspects of Lipschitz geometry. $\endgroup$ – YCor Jan 20 at 18:27

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.