Isomorphism of cosymplectomorphism groups implies diffeomorphism of manifolds

We prove that if two closed, connected, regular cosymplectic manifolds have isomorphic groups of cosymplectomorphisms (as topological groups), then the underlying manifolds are diffeomorphic. The proof proceeds by characterizing the Reeb flow as the center of the group and descending the isomorphism to the symplectic base manifolds. We show that the isomorphism preserves the conjugacy class of the monodromy of the mapping torus, which ensures that the bundle structures, and thus the total spaces are equivalent.

💡 Research Summary

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The paper establishes a striking rigidity theorem for closed, connected, regular cosymplectic manifolds. A regular cosymplectic manifold ((M,\eta,\omega)) of dimension (2n+1) possesses a closed 1‑form (\eta), a closed 2‑form (\omega), and a Reeb vector field (\xi) defined by (\eta(\xi)=1) and (\iota_\xi\omega=0). Regularity means that the Reeb flow is periodic, so the manifold is a principal (S^1)-bundle (\pi:M\to B) over a compact symplectic base ((B,\bar\omega)).

The main result (Theorem 1) states: if two such manifolds (M_1) and (M_2) have isomorphic cosymplectomorphism groups (\mathrm{Cosymp}(M_i)) as topological groups (with the (C^\infty) compact‑open topology), then the underlying manifolds are diffeomorphic, and indeed the cosymplectic structures are equivalent up to a constant scaling of (\omega) and a basic exact adjustment of (\eta).

The proof proceeds in three conceptual steps:

1. Reeb flow as the group centre.
   The identity component (\mathrm{Cosymp}_0(M)) has a centre consisting precisely of the one‑parameter subgroup generated by the Reeb flow. This is shown by examining the Lie algebra: any central element must be a multiple of (\xi) because it must commute with all lifts of Hamiltonian vector fields from the base, which form a centre‑free Lie algebra. Consequently, any topological group isomorphism (\Phi:\mathrm{Cosymp}(M_1)\to\mathrm{Cosymp}(M_2)) maps the Reeb circle in (M_1) onto the Reeb circle in (M_2), preserving periodicity (Proposition 2).

2. Descent to the symplectic base.
   Because the Reeb direction is preserved, each cosymplectomorphism descends to a symplectomorphism of the base. There is an exact sequence
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