Fredholm symplectic structures (Q1358678)

Recall that on a separable real Hilbert space \(E\), there is a bijection \( \Omega \rightarrow \omega:=\langle \Omega \cdot,\cdot\rangle\) between the space \( \mathcal{A}( E) \) of bounded skew symmetric operators on \(E\) and the space \(\Lambda^{2}( E) \) of continuous alternating bilinear forms on \(E\). Identifying operators and forms under this correspondence, we call \(\omega \) nondegenerate (respectively, Fredholm) when \(\Omega \) is invertible (respectively, Fredholm, i.e., with closed image and finite dimensional kernel \(\ker \omega:=\ker \Omega \)) in \(\mathcal{A}( E) \). Generalizing the concept of a symplectic manifold, we call a Hilbert manifold \(M\) (locally modelled on \(E\)) a symplectic Hilbert manifold when it is endowed with a closed (alternating) 2-form \(\widehat{\omega}\), called the symplectic form, which is nondegenerate at each \(x\in M\). This paper is concerned with the `Hamiltonian dynamics' on (connected) submanifolds \(M\) of finite codimension in a symplectic Hilbert manifold \(( W,\widehat{\omega}) \). Since in general, the restriction of the symplectic form \(\widehat{ \omega}\) of \(W\) to the tangent bundle \(TM\) of the submanifold \(M\) is no longer nondegenerate everywhere on \(M\), the authors first study the more general concept of Fredholmian symplectic manifold \(( M,\overline{\omega}) \) where \(\overline{\omega}\) is Fredholm (but not necessarily nondegenerate) at each \(x\in M\). Using the (finite) dimension of \(\ker \omega \), a natural stratification of the fibre bundle \(F^{2}( M) \) of Fredholm alternating bilinear forms \(\omega \) on the tangent spaces of \( M\) is constructed, and the cross sections \(\overline{\omega}\) of \(F^{2}( M) \) that are transversal to the stratum passing through \(\overline{\omega} _{x}\) at each \(x\in M\) are classified. It is shown that embeddings \( i:M\to W\) of a manifold \(M\) as a submanifold of finite (even) codimension in a symplectic Hilbert manifold \(( W,\widehat{\omega}) \) give rise to, generically, Fredholmian symplectic manifolds \(( M,\overline{ \omega}) \) with \(\overline{\omega}:=i^{\ast }\widehat{\omega}\) transversal to the stratification of \(F^{2}( M) \). For such a generic embedding, equivalence is established among the three conditions for a 1-form \(\alpha \) on \(M\): (1) \(\alpha \) has a Hamiltonian vector field \(X_{\alpha }\) over \(M\), i.e., \(\alpha _{x}=\overline{\omega}_{x}( X_{\alpha }( x),\cdot) \) at each \(x\in M\), (2) \(\ker \overline{\omega}_{x}\subset \ker \alpha _{x}\) at every smooth point \(x\) of the critical locus \(\Sigma ( \overline{ \omega}):=\{ x\in M:\ker \overline{\omega}_{x}\neq 0\} \), and (3) \(\alpha \) can be extended to a 1-form \(\widehat{\alpha}\) on \(W\) with a Hamiltonian vector field \(X_{\widehat{\alpha}}\) over \(W\) which is tangent to \(M\).