Moser's Isotopy method and Darboux theorem

1. Symplectic geometry does not exist
2. Isotopy method
- 2.1. Moser's trick
  - - 2.1.0.1. Construction of \(\psi_t\).
- 2.2. Application: Darboux theorem and Moser Stability

The PDF version of this page can be downloaded by replacing html in the its address by pdf. For example /html/sheaf-cohomology.html should become /pdf/sheaf-cohomology.pdf.

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This post is a part of the memoire of my M1 internship at I2M. The memoire contains, needless to say, less errors than this page.

1 Symplectic geometry does not exist

We will prove a symplectic manifold, i.e. a smooth manifold equiped with a closed everywhere non-degenerate 2-form, does not have local invariant. This is a significant difference between symplectic manifold and riemannian manifold, whose local invariance is the curvature.

To do this, one uses a trick of Moser which in the compact case show that isotopic symplectic structures \(\omega_0\) and \(\omega_1\) are strongly isotopic, i.e. \(\psi_t^* \omega_t = \omega_0\)

2 Isotopy method

2.1 Moser's trick

Let \(M\) be a closed manifold (compact, without boundary) and \(\omega_t\) is a family of symplectic structures on \(M\) such that \[ \frac{d}{dt}\omega_t = d\sigma_t \] then there exists diffeomorphism \(\psi_t\) of \(M\) such that \(\psi_t^* \omega_t = \omega_0\)

2.1.0.1 Construction of \(\psi_t\).

One constructs \(\psi_t\) by its flow \(\frac{d}{dt}\psi_t = X_t \circ \psi_t\) such that \[ 0 = \frac{d}{dt}\psi_t^*\omega_t = \psi_t^* \left(\frac{d}{dt} \omega_t + \mathcal{L}_{X_t}\omega_t\right) = \psi_t^* \left( d\sigma_t + X_t \neg d\omega_t + d(X_t \neg \omega_t) \right) \]

Since \(\omega_t\) are closed and non-degenerate, it suffits to choose \(X_t\), which exists uniquely, such that \(X_t \neg \omega = \sigma_t\).

2.2 Application: Darboux theorem and Moser Stability

Using this trick, we can prove the following theorem of Darboux.

Let \(M\) be a closed manifold with symplectic structures \(\omega_0\) and \(\omega_1\) such that they coincide on a fiber \(T_qM\). Then there exists neighborhoods \(\mathcal{N}_0, \mathcal{N}_1\) of \(q\) and a diffeomorphism \(\psi: \mathcal{N}_0 \longrightarrow \mathcal{N}_1\) such that \(\psi^* \omega_1 = \omega_0\).

We remark that it is enough to prove that there exists \(\sigma\) locally defined near \(q\) with \(\omega_1 -\omega_0 = d\sigma\) where \(\sigma = 0\) on \(T_qM\). In fact, let \(\omega_t = \omega_0 + t(\omega_1 - \omega_0)\) one then has a neighborhood \( \mathcal{N}_0\) of \(q\) such that \(\omega_t\) are non-degenerate and the field \(X_t\) constructed by Moser's technique (\(X_t=0\) at \(q\)) has its flow well-defined at time \(t=1\) when starting at \( \mathcal{N}_0\). Then \(\psi_1\) and \( \mathcal{N}_1\) is what we want.

One then uses another trick to construct \(\sigma\): Take any Riemannian metric on \(M\) and let \(\phi_t\) be constructed using the geodesic flow and retricting \( \mathcal{N}_0\) to a geodesic ball such that \(\phi_0|_{\mathcal{N}_0} = q\), \(\phi_1 = Id\) and \(d\phi_t (q) = Id_{T_qM}\). Then

\[ \omega_1 - \omega_0 = \int_0^1 \frac{d}{dt}\phi^*_t(\omega_1 - \omega_0) dt = \int_0^1 \phi_t^* d(Y_t \neg(\omega_1 - \omega_0) )= d \left( \int_0^1 dt \phi_t^*(Y_t \neg (\omega_1 - \omega_0) \right). \]

It is straight-forward to see that the \(\sigma\) constructed this way works.

The theorem of Darboux follows easily from Lemma lem1.

Every two symplectic form \(\omega_0, \omega_1\) on a closed manifold \(M\) are locally isomorphic.