Geometric properties and flux of locally conformally symplectic diffeomorphisms

TL;DR

This work studies the structure of the group of locally conformally symplectic (LCS) diffeomorphisms through S. Haller's LCS flux, exploiting the Hodge decomposition $\omega = dh + l$ to separate exact and non-exact cases. It establishes a short exact sequence $1 \to \operatorname{Ham}_\Omega(M) \to (\ker \Phi)_0 \to H^1_\omega(M)/\Delta \to 1$ and analyzes when it splits topologically, while developing LCS analogues of Weinstein neighborhoods, flux rigidity, Calabi-type invariants, and a unified LCS-Hofer metric. In the non-exact setting, it introduces the Twisted Calabi invariant to capture interactions between Hamiltonian flow and the harmonic Lee component, and confirms these structures in explicit examples such as the Kodaira–Thurston manifold, where a non-Hamiltonian loop and a nonzero Twisted Calabi invariant appear. Overall, the paper extends core symplectic techniques to the LCS realm, clarifying how the harmonic part of the Lee form governs dynamics, topology, and invariants with potential applications to dissipative or conformally twisted systems.

Abstract

We investigate the geometric and topological properties of the group of locally conformally symplectic (LCS) diffeomorphisms, utilizing the LCS flux homomorphism defined by S. Haller. By analyzing the flux map from the universal cover of the identity component $(\ker Φ)_0$ to the first Lichnerowicz cohomology group $H_ω^1(M)$, we establish a short exact sequence characterizing the Hamiltonian subgroup $\Ham_Ω(M)$ and provide conditions for its topological splitting as a semidirect product. We develop LCS analogues of fundamental symplectic results, including a Weinstein neighborhood theorem, a flux rigidity theorem for homotopies, and a characterization of LCS structures on mapping tori. A central theme of this work is the influence of the Hodge decomposition of the Lee form $ω= dh + l$. In the exact case ($l=0$), we utilize the global conformal equivalence to symplectic structures to establish energy-capacity inequalities, an LCS Hofer metric, and non-displaceability results. We explicitly analyze the relationship between the LCS Calabi invariant and its symplectic counterpart, showing they are controlled by a multiplicative factor depending on the conformal weight. For the general non-exact case ($l \neq 0$), we introduce a Twisted Calabi invariant that captures the interaction between Hamiltonian dynamics and the harmonic component of the Lee form.

Theorems & Definitions (67)

• Definition 2.1: LCS Manifold
• Definition 2.2: Morse-Novikov Differential
• Definition 2.3: $d^\omega$-Closed and $d^\omega$-Exact Forms
• Definition 2.4: Lichnerowicz Cohomology
• Remark 2.5: $d^\omega$-Closedness of $\Omega$
• Remark 2.6: Geometric Interpretation
• Proposition 2.7: Lichnerowicz vs. de Rham Cohomology
• proof
• Definition 2.8: Lichnerowicz Codifferential and Laplacian
• Theorem 2.9: Twisted Hodge Decomposition