Local Symplectic Rigidity

DefinitionNo Local Invariants

Darboux's theorem implies that symplectic geometry has no local invariants: any two symplectic manifolds of the same dimension are locally symplectomorphic.

This contrasts with Riemannian geometry, where curvature provides local invariants, and with complex geometry, where holomorphic functions give local structure.

The absence of local invariants is both a simplification and a complication. On one hand, local computations are universal — we can always use Darboux coordinates. On the other hand, all interesting phenomena in symplectic geometry must be global.

ExampleComparison with Riemannian Geometry

For Riemannian manifolds $(M, g)$ and $(M', g')$ :

Local invariant: The Riemann curvature tensor $R_{ijkl}$ distinguishes flat from curved spaces
Example: The sphere $S^2$ and plane $\mathbb{R}^2$ are not locally isometric due to curvature

For symplectic manifolds $(M, \omega)$ and $(M', \omega')$ of the same dimension:

No local invariants: Darboux coordinates make them locally identical
Example: $(S^2, \omega_{\text{sphere}})$ and $(\mathbb{R}^2, \omega_0)$ are locally symplectomorphic

Remark

The slogan is: "Symplectic geometry is the geometry of volume preserving transformations without any infinitesimal structure." All the structure comes from global topology and the requirement that transformations preserve $\omega$ .

Despite local triviality, symplectic manifolds exhibit remarkable global rigidity phenomena discovered in modern symplectic topology:

DefinitionGlobal Rigidity Phenomena

While locally flexible, symplectic structures are globally rigid:

Gromov non-squeezing: Cannot symplectically embed $B^{2n}(r)$ into $B^2(R) \times \mathbb{R}^{2n-2}$ if $r > R$
Arnold conjecture: Hamiltonian symplectomorphisms have many fixed points
Symplectic capacities: Numerical invariants preserved by symplectomorphisms
Lagrangian non-squeezing: Constraints on Lagrangian embeddings

These phenomena have no local counterparts — they depend essentially on global structure.

ExampleNon-Squeezing in Practice

Consider trying to symplectically map a ball of radius 2 in $\mathbb{R}^4$ into a cylinder $B^2(1) \times \mathbb{R}^2$ .

Volume-preserving perspective: This is possible by "squeezing" — compress in one direction, expand in another.

Symplectic perspective: Gromov's theorem says this is impossible symplectically. The symplectic structure provides a global obstruction not visible in any local neighborhood.

DefinitionFlexibility vs Rigidity

Symplectic geometry exhibits a duality:

Flexibility (h-principle):

Isotropic embeddings satisfy an h-principle
Many geometric problems reduce to pure topology

Rigidity (hard analysis):

Symplectic embeddings have constraints
Pseudoholomorphic curves, Floer homology provide obstructions

The boundary between flexible and rigid is an active research area.

Remark

The local-to-global principle in symplectic geometry:

Local: Darboux theorem + Moser method make everything standard
Global: Topology + symplectic invariants (capacities, Floer homology) provide structure

This makes symplectic geometry fundamentally different from differential geometry where local curvature controls global behavior.

ExampleSymplectic Embedding Problem

The symplectic embedding problem asks: when can $(M, \omega)$ be symplectically embedded into $(M', \omega')$ ?

For $M = B^{2n}(r)$ and $M' = \mathbb{R}^{2n}$ , this is always possible (by scaling).

For $M = B^{2n}(r)$ and $M' = B^2(R) \times \mathbb{R}^{2n-2}$ , this requires $r \leq R$ by non-squeezing.

The problem combines local flexibility (Darboux coordinates help construct candidates) with global rigidity (symplectic capacities provide obstructions).

DefinitionLocal Models

While symplectic manifolds have no local invariants, certain substructures do:

Lagrangian submanifolds: Weinstein neighborhood theorem gives local model $T^*L$
Contact hypersurfaces: Neighborhood theorem gives $\mathbb{R} \times M$ contact structure
Singularities: Local models for degenerate forms (Martinet, etc.)

These substructures break the complete local symmetry of Darboux's theorem.