Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the projection of exact Lagrangians in locally conformally symplectic geometry

Adrien Currier

TL;DR

This work analyzes exact Lagrangian submanifolds in cotangent bundles endowed with an exact locally conformally symplectic (lcs) structure, focusing on the beta-exact framework and the implications for projection to the base manifold. It demonstrates that naive extensions of Abouzaid–Kragh-type results fail in this setting and develops an extension theorem (ThmTAF) that localizes the problem via good polygons and a sequence of controlled extensions, enabling a fiber-wise Liouville analysis. The paper also introduces the Liouville-chord perspective as an lcs counterpart to Reeb chords, and connects Lagrangian projections to jet-space Legendrian geometry through generating functions or their obstructions. Together, these results yield constraints on the Lee class, establish simple-homotopy equivalence under suitable hypotheses, and illuminate how Liouville/Reeb chords persist under lcs-type deformations, with concrete examples illustrating the phenomena.

Abstract

In this paper, we construct examples of exact Lagrangians (of "locally conformally symplectic" type) in cotangent bundles of closed manifolds with locally conformally symplectic (lcs) structures and give conditions under which the projection induces a simple homotopy equivalence between an exact Lagrangian and the $0$-section of the cotangent bundle. This line of questioning leads us to investigate the links between the contact geometry of jet spaces and the lcs geometry of cotangent bundles. Among other things, we will study essential Liouville chords, which seem to be the lcs equivalent to Reeb chords. We will also see how Legendrians in jet spaces are an obstruction to the straightforward adaptation of the Abouzaid-Kragh theorem to lcs geometry.

On the projection of exact Lagrangians in locally conformally symplectic geometry

TL;DR

This work analyzes exact Lagrangian submanifolds in cotangent bundles endowed with an exact locally conformally symplectic (lcs) structure, focusing on the beta-exact framework and the implications for projection to the base manifold. It demonstrates that naive extensions of Abouzaid–Kragh-type results fail in this setting and develops an extension theorem (ThmTAF) that localizes the problem via good polygons and a sequence of controlled extensions, enabling a fiber-wise Liouville analysis. The paper also introduces the Liouville-chord perspective as an lcs counterpart to Reeb chords, and connects Lagrangian projections to jet-space Legendrian geometry through generating functions or their obstructions. Together, these results yield constraints on the Lee class, establish simple-homotopy equivalence under suitable hypotheses, and illuminate how Liouville/Reeb chords persist under lcs-type deformations, with concrete examples illustrating the phenomena.

Abstract

In this paper, we construct examples of exact Lagrangians (of "locally conformally symplectic" type) in cotangent bundles of closed manifolds with locally conformally symplectic (lcs) structures and give conditions under which the projection induces a simple homotopy equivalence between an exact Lagrangian and the -section of the cotangent bundle. This line of questioning leads us to investigate the links between the contact geometry of jet spaces and the lcs geometry of cotangent bundles. Among other things, we will study essential Liouville chords, which seem to be the lcs equivalent to Reeb chords. We will also see how Legendrians in jet spaces are an obstruction to the straightforward adaptation of the Abouzaid-Kragh theorem to lcs geometry.
Paper Structure (21 sections, 23 theorems, 82 equations)

This paper contains 21 sections, 23 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments.
  3. A short overview of $\frak{lcs}$ geometry
  4. Definitions
  5. Links with symplectic and contact
  6. The symplectic perspective.
  7. The contact point of view.
  8. New examples of exact Lagrangian submanifolds
  9. On the topology of the Lee class
  10. An extension theorem
  11. Analysis, first part
  12. Analysis, second part
  13. Proof of the theorem \ref{['ThmTAF']}
  14. Restriction to neighborhoods of a good polygon
  15. A series of extensions
  16. ...and 6 more sections

Key Result

Theorem 1.1

There are two connected closed manifold $M$ and $L$ of dimension $n\neq 3$, a $\beta\in\Omega^1(M)$ closed and an embedding such that $i(L)$ is a $\beta$-exact Lagrangian of $(T^*M,\lambda,\beta)$ for $\lambda$ the canonical Liouville form on $T^*M$ and (by slight abuse of notation, we call $\beta$ the pullback of $\beta$ to $T^*(M)$.)

Theorems & Definitions (66)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 1
  • Definition 2.5
  • ...and 56 more