Symplectic curvature and the Arnold form over an isotropic submanifold
The authors study Hamiltonian systems in a \(2d\)-dimensional symplectic manifold \((M, \omega)\) having an isotropic (but non-Lagrangian) compact connected invariant \(k\)-dimensional submanifold \(\Lambda\), by means of a Darboux atlas over \(\Lambda\). They define a Darboux coordinate chart over \(\Lambda\) as a domain \(U\subset M\) together with a set of smooth real independent functions \((\alpha,p;y, \xi)\) on \(U\) such that \(\alpha= (\alpha_1, \dots, \alpha_k)\), \(p=(p^1, \dots, p^k)\), \(y=(y_1, \dots, y_{d-k})\), \(\xi= (\xi^1, \dots, \xi^{d-k})\), \(U\cap \Lambda\neq \emptyset\), \(p|_\Lambda =0\), \(y|_\Lambda =\xi |_\Lambda =0\) and \(\omega |_\Lambda =dp \wedge d\alpha +d\xi \wedge dy\). Then a Darboux atlas \(A\) over \(\Lambda\) is a collection of Darboux coordinate charts \(\{(U, (\alpha,p; y,\xi))\}\) with covers \(\Lambda\) and at the intersection of two charts \(U\) and \(U'\), the corresponding Darboux coordinates \((\alpha,p; y,\xi)\) and \((\alpha', p'; y',\xi')\) satisfy the relations \(p'=Cp\), \((y'\xi')= (y\xi)T^t\), where \(C= (C_{ij})\) with \(C_{ij} \in C^\infty (U\cap U')\), and \(T\) is a constant symplectic matrix, \(T\in SP (2d-2k;\mathbb{R})\). Every Darboux atlas \(A\) over \(\Lambda\) generates a coisotropic extension \(K\) of \(\Lambda\) which is locally given by the equations \(K=\{p_1= \cdots =p_k= 0\}\) and, conversely, a coisotropic extension of \(\Lambda\) determines a Darboux atlas over \(\Lambda\). Moreover, there exists a Darboux atlas over a given isotropic submanifold \(\Lambda\) if and only if there exists a flat symplectic connection on the normal symplectic bundle \(E\) over \(\Lambda\). In this context, the authors prove: suppose that an isotropic submanifold \(\Lambda\) possesses a coisotropic extension \(N\) with zero symplectic curvature, then there is a Darboux atlas \(A\) over \(\Lambda\) such that the coisotropic extension corresponding to \(A\) coincides with \(N\). Conversely, any coisotropic extension of \(\Lambda\) induced by a Darboux atlas has a zero symplectic curvature.
scientific article; zbMATH DE number 1017604
scientific article