Cosymplectic Geometry: Theory & Applications

• Cosymplectic geometry is the study of odd-dimensional manifolds equipped with a closed 1-form and a closed 2-form that together generalize symplectic and contact structures.
• It underpins time-dependent mechanics by introducing canonical coordinates and a unique Reeb vector field, providing a robust framework for integrable systems and field theories.
• Its applications extend from classical dynamics to quantum field theory, influencing manifold topology through cohomological constraints and advanced reduction techniques.

Cosymplectic geometry is the paper of manifolds equipped with a geometrical structure that simultaneously generalizes aspects of symplectic and contact geometry to odd-dimensional settings. A rich and unifying framework, cosymplectic geometry underpins the mathematical description of time-dependent mechanics, the topology of odd-dimensional manifolds, and links to generalized and non-commutative geometric structures. The field encompasses manifold theory, algebraic structures, differential equations, integrable systems, and applications to classical and quantum field theories.

1. Foundational Definitions and Local Structure

A cosymplectic manifold is a smooth (2n+1)-dimensional manifold M endowed with a closed 1-form η and a closed 2-form ω such that the nondegeneracy condition

$\eta \wedge \omega^n \neq 0$

everywhere on M holds (1305.3704). This implies that the pair (η, ω) defines a volume form on M, and that the structure admits a unique Reeb vector field ξ determined by

$i_\xi \omega = 0, \quad \eta(\xi) = 1.$

The canonical local model for cosymplectic geometry follows from a Darboux-type theorem: around any point, there exist coordinates $(t, q^1, ..., q^n, p_1, ..., p_n)$ such that

$$\omega = \sum_{i=1}^n dq^i \wedge dp_i, \quad \eta = dt, \quad \xi = \frac{\partial}{\partial t}.$$

Cosymplectic structures generalize the product of a symplectic manifold with a line or circle, and the formalism is central in the geometric setting of time-dependent Hamiltonian systems.

A closely related class is that of coKähler manifolds, which carry, in addition to (η, ω), an almost contact metric structure (φ, ξ, η, g) satisfying

$$\varphi^2 = -I + \eta \otimes \xi,\quad g(\varphi X, \varphi Y) = g(X, Y) - \eta(X)\eta(Y),$$

with normality and the closedness of η and the fundamental 2-form Φ ensuring integrability (1305.3704).

2. Topological and Cohomological Properties

Cosymplectic geometry imposes strong topological constraints on underlying manifolds. On compact cosymplectic manifolds, the closedness of η implies that the first Betti number $b_1$ is nonzero (1504.02451). Specifically, for coKähler (and K-cosymplectic) manifolds, $b_1$ is always odd, and there exists a splitting of the de Rham cohomology (1504.02451),

$$H^k(M) \cong H^k(M; \mathcal{F}_\xi) \oplus [\eta] \wedge H^{k-1}(M; \mathcal{F}_\xi),$$

where $\mathcal{F}_\xi$ is the foliation generated by ξ. Betti numbers further satisfy monotonicity relations, and in some cases, the Lefschetz map can be constructed on invariant forms, resulting in Lefschetz-type properties akin to those found in Kähler geometry.

For certain cosymplectic structures, the Lefschetz property—a core feature in symplectic geometry—requires restriction to subalgebras invariant under the Reeb flow to guarantee that the Lefschetz map preserves closed forms (1504.02451). Importantly, formality (vanishing higher Massey products in rational homotopy) holds automatically in the coKähler case, but not in general cosymplectic settings.

On 3-manifolds, cosymplectic structures have further significance within the framework of generalized geometry. Every closed oriented 3-manifold supports a B₃-generalized complex structure, which is generically cosymplectic except along codimension-two loci (Porti et al., 19 Feb 2024).

3. Cosymplectic Geometry in Dynamics and Field Theory

Cosymplectic geometry serves as the fundamental geometric context for time-dependent Lagrangian and Hamiltonian systems. In classical mechanics, the phase space for time-dependent systems is modeled on the first jet bundle $J^1\pi$ or $T^*Q \times \mathbb{R}$, with the canonical cosymplectic forms

$$\eta = dt,\qquad \omega = \sum_{i=1}^n dq^i \wedge dp_i,$$

dictating the equations of motion (1305.3704).

Hamiltonian vector fields are defined as

$i_{X_f} \omega = df - \xi(f)\eta,$

leading to a Poisson bracket structure $\{f, g\} = \omega(X_f, X_g)$. The cosmic role of the Reeb vector field in time-dependent mechanics is prominent: the evolution vector field is given by

$E_H = \xi + X_H,$

where $X_H$ is the Hamiltonian vector field for H (Lucas et al., 2023). The geometric Hamilton–Jacobi equation on a cosymplectic manifold for a generating function W reads

$$\frac{\partial W}{\partial t} + H(q, \frac{\partial W}{\partial q}, t) = 0,$$

providing a method for integrating time-dependent systems by lifting solutions from configuration space to phase space (1612.06224).

The cosymplectic framework extends naturally to first-order classical field theories via k-cosymplectic geometry (1409.5604). Here, the phase space is equipped with k closed 1-forms and k closed 2-forms, allowing simultaneous treatment of multiple independent variables (e.g., spacetime coordinates). The corresponding field equations can be written in a coordinate-free way, and the framework is closely related to multisymplectic formalism.

A defining advantage in cosymplectic reduction—the cosymplectic Marsden–Weinstein process—is that the presence of the closed 1-form $\eta$ allows time to be retained as a coordinate during reduction. The reduced space inherits a cosymplectic structure, and the interplay between groupoid and algebroid reduction commutes, mirroring phenomena seen in Poisson geometry (Garcia et al., 5 Mar 2024).

4. Advanced Structures and Generalizations

Cosymplectic geometry extends to incorporate multi-structures, algebraic analogues, and generalized frameworks:

4.1. Three-cosymplectic Structures

A 3-cosymplectic manifold is of dimension $4n+3$ and is equipped with three almost cosymplectic structures $(\varphi_\alpha, \xi_\alpha, \eta_\alpha)$, $\alpha = 1,2,3$, linked by quaternionic-type relations (1209.1490). These structures satisfy

$$\varphi_\gamma = \varphi_\alpha \varphi_\beta - \eta_\beta \otimes \xi_\alpha = -\varphi_\beta \varphi_\alpha + \eta_\alpha \otimes \xi_\beta$$

for even permutations, and enjoy parallelism and total geodesy properties. The basic cohomology decomposes in a way that imposes stringent topological restrictions, and the space of harmonic forms inherits an so(4,1) action, revealing deep symmetries with implications for representation theory and theoretical physics.

4.2. Structures on Algebraic Systems

A cosymplectic Jacobi–Jordan algebra is an odd-dimensional algebra (g, ·) equipped with a 1-form α and a skew-symmetric bilinear form ω such that $\alpha \wedge \omega^n \neq 0$, α(x·y) = 0, and a specific Jacobiator vanishes (Bourkadi et al., 29 Apr 2024). These structures have a one-to-one correspondence with symplectic JJ-algebras of even dimension equipped with an anti-derivation and support a right-skew-symmetric product, generalizing the classical symplectic construction.

4.3. Co-complex and Lagrangian-like Subspaces

Cosymplectic analogues of symplectic Lagrangian subspaces, termed Lagrangian-like subspaces, are those isotropic subspaces not containing the Reeb vector, with cosymplectic annihilators augmenting by the Reeb direction. The Lagrangian Grassmannian in this context forms a homogeneous space linked to moduli questions and local-to-global compatibility (Tchuiaga et al., 1 Jan 2025).

Co-complex structures, acting as cosymplectic analogues of complex structures, are compatible endomorphisms satisfying specific splitting and positivity conditions (e.g., mapping ker η isotropically), with the set of such structures being contractible (Tchuiaga et al., 1 Jan 2025).

5. Integrable Systems, Reductions, and Groupoids

Cosymplectic geometry supports a robust theory of integrable systems, action-angle variables, and reduction procedures:

• Integrability: Systems generated by the evaluation vector field $Y_H = \xi + X_H$ are integrable if there exist enough first integrals (with certain commutation relations), and the action-angle theorem holds on invariant tori, with canonical forms expressed in extended coordinates (Jovanovic et al., 2022).
• Reduction: Cosymplectic reduction processes, both on manifolds and groupoids, parallel the Marsden-Weinstein method but are adapted to odd dimensions and include a closed 1-form η in all structural considerations. Reduction at the groupoid and Lie algebroid levels commutes, ensuring structural compatibility (Garcia et al., 5 Mar 2024).
• Coisotropic Embeddings: Every precosymplectic manifold can be coisotropically embedded into a cosymplectic manifold, extending Gotay’s theorem; locally unique cosymplectic extensions and local normal forms via Darboux coordinates facilitate analysis of constrained time-dependent systems (León et al., 20 Oct 2024).
• Generalized Structures: Cosymplectic structures provide the archetype for stable B_n-generalized complex structures in odd dimensions, and surgery techniques demonstrate that every closed orientable 3-manifold supports generalized structures that are generically cosymplectic (Porti et al., 19 Feb 2024).

6. Concrete Constructions, Submanifolds, and Applications

Cosymplectic geometry provides explicit models and sharp results in several domains:

• Cosymplectic p-spheres and Circles: Linear combinations of closed cosymplectic pairs yield higher-dimensional spheres of structures (p-spheres). In dimension three, a cosymplectic circle is taut if and only if it is round, leading to a classification of admitting manifolds (3-torus or certain nilmanifolds), and taut cosymplectic circles canonically induce complex structures on products M × ℝ (1406.2242).
• Submanifolds in Cosymplectic Geometry: Various classes of submanifolds (e.g., CRS bi-warped products) satisfy sharp curvature inequalities tying the second fundamental form to the geometry of the constituent factors, with rigidity and energy estimates for warped products, and explicit realizations in Euclidean spaces (Chen et al., 2018).
• Dynamics and Universal Computation: Cosymplectic geometry enables the construction of stationary, Turing complete solutions to the Navier–Stokes equations on 3-manifolds with non-vanishing harmonic 1-forms. Such manifolds admit cosymplectic structures where the Reeb dynamics afford programmable computational behavior, generalizing the correspondence between Beltrami fields and Reeb flows to viscous fluids (Dyhr et al., 10 Jul 2025).
• Constraints in Field Theory: The k-cosymplectic formalism and its extension to k-precosymplectic structures allow for a constraint algorithm resolving singular Lagrangian and Hamiltonian systems in classical field theory (Gràcia et al., 2018).

7. Symmetries, Diffeomorphism Groups, and Quantitative Geometry

The structure of transformation groups preserving cosymplectic data underpins much of the field’s rigidity and flexibility:

• Cosymplectic Diffeomorphisms: The group of diffeomorphisms preserving both ω and η (or preserving ω and η modulo scaling) forms the basis for cosymplectic and almost cosymplectic transformation groups. The Reeb field’s invariance under these maps is a key rigidity criterion (Tchuiaga et al., 2019).
• Flux and Norms: The cosymplectic flux homomorphism quantifies the deformation of structures under isotopy; bi-invariant and Hofer-like norms measure the “size” of cosymplectic transformations, providing analogs to symplectic capacity and nondegeneracy (Tchuiaga et al., 2019, Tchuiaga et al., 1 Jan 2025).
• Deformation Theory: Both infinitesimal and global deformation methods (including Moser-type isotopies, tubular neighborhood theorems, homothetic deformations, and double extensions in algebra) play crucial roles in analyzing the moduli of cosymplectic structures (Tchuiaga et al., 1 Jan 2025, Bourkadi et al., 29 Apr 2024).

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Cosymplectic geometry, integrating elements from symplectic, contact, and complex geometry, establishes a foundational framework for the analysis of odd-dimensional manifolds, time-dependent mechanical systems, and algebraic structures. Its depth is reflected in strong topological consequences, rich deformation theory, applications to mathematical physics, and unexpected connections with universality in dynamical systems.