Russian Mathematical Surveys, 2025, Volume 80, Issue 1, Pages 153–154 DOI: https://doi.org/10.4213/rm10210e (Mi rm10210)

DOI: https://doi.org/10.4213/rm10210e

Funding agency	Grant number
Russian Science Foundation	22-11-00299
This work was supported by the Russian Science Foundation under grant no. 22-11-00299, https://rscf.ru/en/project/22-11-00299/.

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A 2-plane distribution on a smooth 3-manifold is called a contact structure if locally it is the kernel of a smooth differential 1-form $\alpha$ such that $\alpha\wedge d\alpha\neq0$ everywhere. Given a piecewise smooth submanifold of a contact manifold, how can one define for two smoothings of it their equivalence to each other and to the original submanifold? Two smooth submanifolds can be said to be equivalent if a diffeomorphism of the ambient manifold exists that maps one submanifold to the other while preserving the contact structure. A diffeomorphism with the latter property is called contact. To define the equivalence of piecewise smooth submanifolds, it is necessary to extend the group of contact diffeomorphisms. Similarly, one can ask how to generalize the concept of Legendrian isotopy so that it be applicable to piecewise smooth submanifolds. We consider the Lipschitz structure on a manifold induced by the smooth structure. We consider natural generalizations of the concepts of a Legendrian curve and a Legendrian isotopy to bi-Lipschitz locally flat submanifolds, and we show that Legendrian equivalence classes are the same as in the smooth case. Also for homeomorphisms preserving the Lipschitz structure we define what it means to be contact, and we define contact isotopies.

We assume that the manifold is equipped with a Riemannian metric and is connected. The Riemannian metric induces a path metric on the manifold such that the distance between two points equals the infimum of the length of a path connecting these points. In what follows nothing will depend on the choice of the Riemannian metric, since the restrictions of the path metrics induced by two arbitrary Riemannian metrics to any compact subset are comparable. A rectifiable curve is called $C$-Lavrentiev if for any pair of its points there is an arc of this curve connecting them such that the ratio of the length of the arc to the distance between its endpoints does not exceed $C.$ A curve is called Lavrentiev if it is $C$-Lavrentiev for some $C.$ This property was considered by Lavrentiev in [1]. A rectifiable curve is called Legendrian if the Riemann–Stieltjes integral of the contact form over any compact subarc is zero. This definition is independent of the particular choice of a contact form $\alpha$ which locally defines the contact structure.

Let $X$ and $Y$ be metric spaces. Recall that a map $f\colon X\to Y$ is called $C$-bi-Lipschitz if for any two points $x,y\in X$

A map is called bi-Lipschitz if it is $C$-bi-Lipschitz for some $C$. A submanifold $L\subset M$ of dimension $q$ is called topologically (respectively, bi-Lipschitz) locally flat if for any point $p\in L$ there is a neighbourhood $U\subset M$ of $p$, an open subset $W$ of Euclidean space, and a homeomorphism (respectively, a bi-Lipschitz homeomorphism) $\varphi\colon U \to W$ such that $\varphi(L\cap U) = \mathbb R^q\cap W,$ where $\mathbb R^q$ is a $q$-dimensional subspace. A Lavrentiev curve in a three-dimensional manifold is not necessarily topologically flat. Furthermore, even a topologically flat Lavrentiev curve is not necessarily bi-Lipschitz locally flat: see an example in [2]. However, Legendrian Lavrentiev curves are locally flat as the following proposition shows.

A one-dimensional compact submanifold without boundary will be called a link. A family of maps is called uniformly bi-Lipschitz if there exists a number $C$ such that all these maps are $C$-bi-Lipschitz. Legendrian Lavrentiev links $L_0$ and $ L_1$ in a contact manifold $M$ are called Legendrian equivalent if there exists a uniformly bi-Lipschitz continuous family of maps $f_t\colon L_0 \to M,$ where $t\in[0;1]$, such that $f_0 = \mathrm{id}_{L_0},$ the link $f_t(L_0)$ is Legendrian for any $t\in[0;1]$, and $f_1(L_0) = L_1.$ The family $\{f_t\}_{t\in[0;1]}$ is called a Legendrian isotopy from $L_0$ to $L_1.$

Theorem 2 is only the first step towards constructing piecewise smooth contact topology in dimension 3. As mentioned at the beginning, we also need to extend the group of contact diffeomorphisms. A map $f\colon X\to Y$ is called locally bi-Lipschitz if each point $x\in X$ lies in some neighbourhood such that the restriction of the map to this neighbourhood is bi-Lipschitz. A locally bi-Lipschitz homeomorphism is called contact if it preserves the class of Legendrian Lavrentiev links. Let us also give a definition of a contact isotopy. A family of maps is called locally uniformly bi-Lipschitz if for each point in the source space there is a neighbourhood of this point such that the restriction of the family to this neighbourhood is uniformly bi-Lipschitz. A contact ambient isotopy is a continuous locally uniformly bi-Lipschitz ambient isotopy in the class of contact locally bi-Lipschitz homeomorphisms. Proposition 3 follows directly from the definitions.

In the smooth case the converse is also true: any Legendrian isotopy extends to an ambient contact isotopy. For Lavrentiev links this is unknown. It is also unknown whether the weaker claim is true: can any Legendrian Lavrentiev link be mapped to a smooth one by a contact locally bi-Lipschitz homeomorphism?

The author is grateful to I. A. Dynnikov for stating and discussing the problem.

Bibliography
1.	M. A. Lavrentiev, Mat. Sb., 1(43):6 (1936), 815–846 (Russian)
2.	P. Tukia, Ann. Acad. Sci. Fenn. Ser. A I Math., 5:1 (1980), 49–72

Citation in format AMSBIB

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\paper Legendrian Lavrentiev links
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\yr 2025
\vol 80
\issue 1
\pages 153--154
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