Symplectic isotopy - Knowlpedia

Definition

A symplectic isotopy of a symplectic manifold $(M,\omega)$ is a smooth path

\phi_t\in\operatorname{Symp}(M,\omega),\qquad 0\leq t\leq1,

with $\phi_0=\operatorname{id}_M$ . Its time-dependent generating vector field $X_t$ is defined by

\frac{d}{dt}\phi_t=X_t\circ\phi_t.

Differentiating $\phi_t^*\omega=\omega$ gives

0=\phi_t^*(\mathcal L_{X_t}\omega) =\phi_t^*d(\iota_{X_t}\omega).

Hence each one-form

\alpha_t=\iota_{X_t}\omega

is closed. Conversely, a time-dependent vector field with closed $\alpha_t$ generates a symplectic isotopy wherever its flow exists.

Hamiltonian versus symplectic

The isotopy is Hamiltonian when the closed forms $\alpha_t$ are exact and may be written

\alpha_t=dH_t

for a smooth time-dependent Hamiltonian. Thus Hamiltonian isotopies are symplectic, but the converse can fail when $H^1_{\mathrm{dR}}(M)\neq0$ . The flux class

\left[\int_0^1\alpha_t\,dt\right]\in H^1_{\mathrm{dR}}(M)

records the integrated cohomological obstruction.

Support convention

On a closed manifold no support condition is needed. On a noncompact manifold, a compactly supported symplectic isotopy means that the generating fields $X_t$ have support in one fixed compact subset of $M$ for all $t$ . Equivalently, the isotopy is stationary outside that compact set. Its contraction forms are compactly supported and naturally define compactly supported cohomology classes.

This convention concerns the whole path, not only its endpoint: a compactly supported endpoint may admit paths with different support behavior.

References

Dusa McDuff and Dietmar Salamon, Introduction to Symplectic Topology, 3rd ed., Oxford University Press, 2017. DOI record. Relevant: §10.2, symplectic isotopies and flux.
Augustin Banyaga, The Structure of Classical Diffeomorphism Groups, Kluwer Academic Publishers, 1997. DOI record. Relevant: Chapter 7.