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Symplectomorphism

A symplectomorphism is a diffeomorphism \phi: (M_1, \omega_1) \to (M_2, \omega_2) between two symplectic manifolds that preserves the symplectic forms, meaning \phi^*\omega_2 = \omega_1.^[1] This preservation ensures that the nondegenerate, closed 2-form structure defining the symplectic geometry is maintained under the map.^[2] In symplectic geometry, symplectomorphisms serve as the structure-preserving maps between symplectic manifolds, analogous to isometries in Riemannian geometry, and they form the symplectomorphism group \mathrm{Symp}(M, \omega) on a given manifold.^[1] A key property is that every symplectic manifold is locally symplectomorphic to the standard symplectic space (\mathbb{R}^{2n}, \omega_0 = \sum_{i=1}^n dx_i \wedge dy_i) by the Darboux theorem, highlighting the uniformity of local structure despite global variations.^[1] The graphs of symplectomorphisms are Lagrangian submanifolds, which are maximal isotropic subspaces with respect to the symplectic form, underscoring their role in studying intersections and fixed points.^[1] On compact manifolds with vanishing first de Rham cohomology, symplectomorphisms close to the identity possess at least two fixed points, as per extensions of the Arnold conjecture.^[1] Symplectomorphisms are fundamental in Hamiltonian mechanics, where they correspond to canonical transformations that preserve the phase space structure and Poisson brackets \{f, g\} = \omega(X_f, X_g).^[2] The time evolution of a Hamiltonian system generates a one-parameter group of symplectomorphisms via the flow of the Hamiltonian vector field X_H, defined by \omega(X_H, \cdot) = -dH, ensuring conservation of the symplectic volume and Liouville's theorem.^[1] In broader applications, they facilitate symplectic reduction and moment maps for symmetries, enabling the analysis of conserved quantities and equivariant structures in dynamical systems.^[1]

Definition and Basics

Formal Definition

A symplectic manifold is an even-dimensional smooth manifold M equipped with a symplectic form \omega, which is a closed non-degenerate 2-form on M.^[1] The closedness condition requires that the exterior derivative vanishes, d\omega = 0, ensuring that \omega defines a presymplectic structure that is compatible with the manifold's topology.^[1] Non-degeneracy means that for every point p \in M, the map \tilde{\omega}_p: T_p M \to T_p^* M given by \tilde{\omega}_p(v)(u) = \omega_p(v, u) is a linear isomorphism, implying that \omega_p pairs tangent vectors with cotangent vectors bijectively.^[1] A symplectomorphism is a diffeomorphism f: (M, \omega) \to (N, \omega') between two symplectic manifolds that preserves the symplectic structure, satisfying f^* \omega' = \omega.^[1] The pullback operation f^* \omega' is defined pointwise by (f^* \omega')_p(u, v) = \omega'_{f(p)}(df_p(u), df_p(v)) for p \in M and tangent vectors u, v \in T_p M, which ensures that the symplectic form on N is transported back to match \omega on M.^[1] In local Darboux coordinates (q_1, \dots, q_n, p_1, \dots, p_n) on M and similar coordinates on N, where \omega = \sum_{i=1}^n dq_i \wedge dp_i and \omega' = \sum_{i=1}^n dq_i' \wedge dp_i', the condition f^* \omega' = \omega translates to the Jacobian matrix J = df_p satisfying J^T \Omega J = \Omega, with \Omega = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix} the standard symplectic matrix.^[3] Symplectomorphisms are distinguished as global or local depending on their domain: a global symplectomorphism is defined on the entire manifolds M and N, while a local symplectomorphism is a diffeomorphism between open subsets U \subset M and V \subset N satisfying the pullback condition on those sets, though it may not extend to the whole manifolds due to topological obstructions.^[1]

Examples

In classical mechanics, symplectomorphisms appear as canonical transformations on the phase space \mathbb{R}^{2n}, endowed with the standard symplectic form \omega = \sum_{i=1}^n dq_i \wedge dp_i. These transformations map old canonical coordinates (q_i, p_i) to new ones (Q_i, P_i) while preserving the structure of Hamilton's equations, meaning the transformed Hamiltonian system retains the same form. Equivalently, a diffeomorphism \phi: \mathbb{R}^{2n} \to \mathbb{R}^{2n} is canonical if it satisfies \phi^* \omega = \omega, ensuring the symplectic structure is invariant.^[4] Linear symplectomorphisms on \mathbb{R}^{2n} form the symplectic group Sp(2n, \mathbb{R}), consisting of all $2n \times 2n invertible real matrices A that preserve the standard symplectic form, satisfying A^T J A = J where J = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix}. These matrices represent linear changes of coordinates in phase space that maintain the symplectic structure, such as rotations in the (q_i, p_i)-planes combined with appropriate adjustments in conjugate variables. For instance, in n=1, elements of Sp(2, \mathbb{R}) include matrices like \begin{pmatrix} a & b \\ c & d \end{pmatrix} with ad - bc = 1, preserving areas in the (q, p)-plane.^[1] In two dimensions, symplectomorphisms simplify to area-preserving diffeomorphisms. On \mathbb{R}^2 with \omega = dq \wedge dp, any diffeomorphism preserving this form locally preserves areas, as \omega defines the standard area measure. Similarly, on the two-dimensional torus T^2, equipped with a compatible area form, symplectomorphisms are orientation-preserving diffeomorphisms that conserve the total area, playing a key role in studies of dynamical systems like twist maps. Examples include the standard map on T^2, which models chirikov's standard map in Hamiltonian chaos while preserving the symplectic area. A concrete example of a nonlinear symplectomorphism on \mathbb{R}^{2n} arises from coordinate changes that preserve the symplectic structure, such as the transformation (q_i, p_i) \mapsto (q_i + f_i(p), p_i) for smooth functions f_i: \mathbb{R}^n \to \mathbb{R}. For n=1, the map (q, p) \mapsto (q + f(p), p) satisfies \phi^* \omega = dq \wedge dp + df \wedge dp = dq \wedge dp, since df \wedge dp = f'(p) \, dp \wedge dp = 0, thus preserving the form. This type of shear transformation is canonical and can simplify Hamiltonians, for example, in action-angle variables.^[5]

Dynamical Properties

Hamiltonian Flows

In symplectic geometry, the Hamiltonian vector field X_H associated to a smooth function H: (M, \omega) \to \mathbb{R} on a symplectic manifold (M, \omega) is defined by the equation \omega(X_H, \cdot) = -dH. This vector field encodes the dynamics of the classical mechanical system governed by the Hamiltonian H, where \omega is the symplectic form. The integral curves of X_H generate a one-parameter group of diffeomorphisms \phi_t: M \to M, known as the Hamiltonian flow, satisfying \frac{d}{dt} \phi_t(p) = X_H(\phi_t(p)) with \phi_0 = \mathrm{id}. This flow preserves the symplectic structure, meaning \phi_t^* \omega = \omega for all t, establishing that each \phi_t is a symplectomorphism. The preservation arises because the Lie derivative of the symplectic form along X_H vanishes: \mathcal{L}_{X_H} \omega = 0. To see this, note that \mathcal{L}_{X_H} \omega = d(i_{X_H} \omega) + i_{X_H} d\omega = d(-dH) + 0 = 0, since d\omega = 0 by the closedness of \omega.^[6] Additionally, the flow conserves energy levels, satisfying H \circ \phi_t = H for all t. This follows from the fact that along trajectories, \frac{d}{dt} (H \circ \phi_t) = dH(X_H) = -[\omega](/page/Omega)(X_H, X_H) = 0, as [\omega](/page/Omega) is skew-symmetric. A key consequence is Liouville's theorem, which states that the Hamiltonian flow preserves the symplectic volume form \frac{\omega^n}{n!} on the $2n-dimensional manifold M, where n is the dimension of the underlying real vector space.^[7] This volume preservation ensures that phase space volumes remain invariant under time evolution, reflecting the incompressibility of Hamiltonian dynamics.^[7]

Group of Symplectomorphisms

The group of symplectomorphisms of a symplectic manifold (M, \omega), denoted \mathrm{Symp}(M, \omega), consists of all diffeomorphisms \phi: M \to M satisfying \phi^*\omega = \omega. This group is equipped with the C^\infty-topology and forms an infinite-dimensional Lie pseudogroup. Within \mathrm{Symp}(M, \omega), the subgroup \mathrm{Ham}(M, \omega) comprises the Hamiltonian symplectomorphisms, which are the time-1 maps of flows generated by Hamiltonian vector fields. This subgroup is normal and path-connected.^[8]^[9] For compact connected symplectic manifolds, Banyaga's theorem asserts that \mathrm{Ham}(M, \omega) is a simple group, meaning it has no nontrivial normal subgroups.^[8] The Hofer metric on \mathrm{Ham}(M, \omega) is defined as the infimum over all Hamiltonians generating a path from the identity to a given element, yielding a complete bi-invariant metric that induces a Finsler geometry on the group.^[10] Not all elements of \mathrm{Symp}(M, \omega) belong to \mathrm{Ham}(M, \omega), particularly on non-compact manifolds where the flux homomorphism detects non-Hamiltonian symplectomorphisms; for instance, on the cylinder S^1 \times \mathbb{R} with the standard symplectic form, certain area-preserving maps isotopic to the identity through symplectomorphisms are not Hamiltonian.^[9]^[11]

Geometric Comparisons

With Riemannian Geometry

In Riemannian geometry, an isometry is a diffeomorphism \phi: (M, g) \to (M', g') that preserves the metric tensor, satisfying g'(\mathrm{d}\phi(X), \mathrm{d}\phi(Y)) = g(X, Y) for all vector fields X, Y. The group of isometries \mathrm{Isom}(M, g) of a compact Riemannian manifold is a finite-dimensional Lie group, typically of dimension at most \frac{1}{2} \dim(M) (\dim(M) + 1). In contrast, symplectomorphisms preserve only the symplectic form \omega, satisfying \phi^* \omega' = \omega, and the group \mathrm{Symp}(M, \omega) of symplectomorphisms of a symplectic manifold is an infinite-dimensional Fréchet Lie group, exhibiting significantly less rigidity than the isometry group.^[12] A representative example illustrates this difference on the 2-sphere S^2. Equipped with the standard round metric, the isometry group is the orthogonal group O(3), a compact 3-dimensional Lie group consisting of rotations and reflections. However, with the standard area form as symplectic structure, the symplectomorphism group is larger, comprising all area-preserving diffeomorphisms, and is infinite-dimensional.^[13]^[12] These structural disparities have profound implications for classification. Riemannian manifolds are locally determined by their curvature tensor, allowing distinction via local invariants like sectional curvature. Symplectic manifolds, however, admit Darboux coordinates locally, with no nontrivial local invariants beyond the dimension, underscoring the greater flexibility in symplectic geometry.

Rigidity and Local Structure

One of the fundamental results in symplectic geometry is the Darboux theorem, which asserts that every symplectic manifold (M, \omega) of dimension $2n is locally symplectomorphic to the standard symplectic space (\mathbb{R}^{2n}, \omega_0), where \omega_0 = \sum_{i=1}^n dq_i \wedge dp_i. This means that around any point in M, there exist local coordinates (q_1, \dots, q_n, p_1, \dots, p_n) such that the symplectic form \omega takes the canonical form \omega_0 in these coordinates. The theorem highlights the local flexibility of symplectic structures, as it implies that symplectic manifolds possess no local invariants analogous to curvature in Riemannian geometry. A related local normalization result is the Moser theorem, which addresses the stability of symplectic forms under deformations. Specifically, if two symplectic forms \omega_0 and \omega_1 on a compact manifold M are isotopic through a smooth path \omega_t (with t \in [0,1]) such that [\omega_t] = [\omega_0] in the de Rham cohomology (i.e., they lie in the same cohomology class), then there exists a diffeomorphism \phi: M \to M isotopic to the identity such that \phi^* \omega_1 = \omega_0. This theorem, often proved using Moser's trick of solving a certain homotopy equation for vector fields, demonstrates that symplectic forms within the same cohomology class are equivalent up to symplectomorphism on compact manifolds. These local theorems have profound implications for the classification of symplectic manifolds. Unlike Riemannian manifolds, where local invariants like sectional curvature provide obstructions to isometry, symplectic manifolds lack such local invariants due to the Darboux theorem, making local classification trivial. However, global topological features, such as the fundamental group or homology, play a crucial role in distinguishing symplectic structures, as the Moser theorem preserves cohomology classes but does not address global embedding or topological obstructions. Despite this local flexibility, symplectic geometry exhibits remarkable global rigidity phenomena, as exemplified by the Gromov width, a symplectic capacity that measures the largest standard ball embeddable into a symplectic manifold while respecting the symplectic structure. The Gromov nonsqueezing theorem establishes that the Gromov width of a symplectic manifold provides a rigid obstruction to embeddings, preventing "squeezing" of higher-dimensional balls into lower-dimensional cylinders symplectically. This rigidity contrasts sharply with the local triviality from Darboux and Moser theorems, underscoring how symplectic invariants emerge globally through techniques like pseudoholomorphic curves.

Advanced Topics

Quantizations

Quantization procedures in symplectic geometry provide a bridge between classical mechanics on symplectic manifolds and quantum mechanics on Hilbert spaces, where symplectomorphisms play a central role by inducing unitary operators that preserve the quantum structure. In this framework, a classical phase space (M, \omega) is quantized to a Hilbert space \mathcal{H}, and a symplectomorphism \phi: M \to M lifts to a unitary operator U_\phi: \mathcal{H} \to \mathcal{H} such that the quantization map intertwines the classical and quantum actions, ensuring that quantum observables evolve according to the classical symplectic dynamics. This correspondence is foundational in geometric quantization, as developed by Kostant and Souriau, where the symplectic form \omega dictates the commutation relations in the quantum algebra.^[14] Prequantization is the initial step, associating to the symplectic manifold (M, \omega) a complex line bundle L \to M equipped with a Hermitian connection whose curvature form equals \omega / \hbar, where \hbar is the reduced Planck's constant. Symplectomorphisms on M that preserve the cohomology class of \omega lift to automorphisms of the prequantum bundle L, preserving the connection and thus the parallel transport, which corresponds to the classical flow in the quantum setting. For Hamiltonian symplectomorphisms generated by a function H, the lift is explicitly given by multiplication by the phase factor \exp(i H / \hbar) on sections of L, ensuring a unitary representation. However, the prequantum Hilbert space L^2(M, L) is typically infinite-dimensional and overcomplete, necessitating further refinement.^[15] Geometric quantization refines prequantization by incorporating a choice of polarization—a maximal positive Lagrangian subbundle of the complexified tangent bundle—and half-forms to correct for the transformation properties under symplectomorphisms. The quantum Hilbert space consists of square-integrable sections of L \otimes K^{1/2}, where K is the canonical bundle, that are holomorphic along the polarization; symplectomorphisms act on these sections via the metaplectic representation, which ensures unitarity up to a phase determined by the Maslov index. This representation arises from the action on half-densities and captures the quantum evolution, with observables quantized as operators on the polarized sections.^[16] A concrete example occurs on the standard symplectic space \mathbb{R}^{2n} with the canonical form \omega = \sum dq_i \wedge dp_i, where the group of linear symplectomorphisms is \mathrm{Sp}(2n, \mathbb{R}), and its double cover is the metaplectic group \mathrm{Mp}(2n, \mathbb{R}). The metaplectic representation provides unitary operators on the quantum Hilbert space L^2(\mathbb{R}^n), faithfully realizing \mathrm{Mp}(2n, \mathbb{R}) and thus \mathrm{Sp}(2n, \mathbb{R}) projectively; for the quantum harmonic oscillator, this action rotates the phase space coordinates while preserving the energy levels, illustrating how classical symplectic transformations quantize to ladder operators.^[17]^[18] Challenges arise because not all symplectomorphisms quantize to unitary operators without anomalies; topological obstructions, such as the non-vanishing of the Maslov class or the failure to lift to the prequantum bundle, prevent a consistent unitary lift for general symplectomorphisms, particularly those not isotopic to the identity on non-simply connected manifolds. These anomalies manifest as phase inconsistencies in the representation, requiring the double cover to resolve them in linear cases but leading to projective rather than true unitary representations in general.^[19]

Arnold Conjecture

The Arnold conjecture, proposed by Vladimir Arnold in 1965, asserts that for a non-degenerate Hamiltonian symplectomorphism \phi on a compact symplectic manifold (M, \omega) of dimension $2n, the number of fixed points of \phi is at least the sum of the Betti numbers of M, that is, \# \operatorname{Fix}(\phi) \geq \sum_{i=0}^{2n} b_i(M).^[20] This lower bound exceeds the Euler characteristic \chi(M) in general and provides a sharp homological estimate for the minimal number of fixed points. The conjecture applies specifically to the Hamiltonian subgroup of the symplectomorphism group, where \phi is the time-1 map of a Hamiltonian flow generated by some smooth time-dependent Hamiltonian H: S^1 \times M \to \mathbb{R}.^[21] The motivation for the conjecture draws from Morse theory, viewing Hamiltonian diffeomorphisms as analogous to gradient flows of Morse functions on M. Fixed points of \phi correspond to critical points of the symplectic action functional on the loop space of M, whose Morse inequalities would imply the desired bound if a suitable infinite-dimensional Morse theory could be developed.^[22] Partial results toward the conjecture include weaker estimates from the Lusternik–Schnirelmann category: for a Hamiltonian symplectomorphism homotopic to the identity, the number of fixed points is at least the LS category of M plus one, which is bounded above by \sum b_i(M) but often strictly smaller. Full proofs exist in low dimensions, such as for surfaces (where the bound is \sum b_i = 2 + 2g for genus g) and tori, using variational methods or holomorphic curve techniques.^[23] A major breakthrough came with the introduction of Hamiltonian Floer homology by Andreas Floer in 1986, which defines a chain complex generated by non-degenerate 1-periodic orbits (fixed points of \phi) and differentials via moduli spaces of pseudoholomorphic cylinders; this yields an isomorphism with the ordinary cohomology of M over suitable Novikov rings, proving the conjecture for semi-positive (including monotone) symplectic manifolds. The theory was later extended to all closed symplectic manifolds without positivity assumptions, using virtual techniques to handle bubbling issues.^[24] The conjecture has profound implications for symplectic topology, establishing minimal numbers of periodic orbits for generic Hamiltonians and, through symplectization, linking to the existence of closed Reeb orbits on associated contact manifolds, as in analogs of the Weinstein conjecture.^[25] As of 2025, the conjecture is proven in many cases, including all monotone symplectic manifolds via Floer homology and its refinements, and fully for general closed symplectic manifolds over the rationals using A_\infty-structures and Gromov–Witten invariants, as independently shown by Fukaya–Ono and Liu–Tian.^[26] Extensions to relative or singular settings continue to leverage these tools, though the conjecture remains open over the integers in full generality.^[27]

References

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