Symplectic leaves

introduction

symplectic geometry
The symplectic leaves are equivalence relations \(x \sim y\) if and only if \(x\) can be connected to \(y\) be a piece-wise Hamiltonian path
Let \(D\) be a degenerate distribution
this means that for every point \(x \in M\), \(D_x\) is a subset of \(T_x M\)
subset = subspace
distribution normally means that \(D_x\) is constant rank
and \(D_x\) is spanned by vector fields
which means that for every \(x\) there is vector fields \(X_1,\ldots,X_r\) locally defined around \(x\) such that \(X_1(x),\ldots,X_r(x)\) span \(D_x\)
and \(X_1(y),\ldots,X_r(y)\) lie in \(D_y\) for all \(y\) where they are defined
a foliation of \(D\) is an immersed manifold \(A\) of \(M\) with \(TA = D\)
Let \(M^{dis}\) be the manifold with underlying set \(M\) and the discrete topology
\(M^{dis}\) is an immersed manifold for \(D = M \times 0\)
\(M = \R^2\)
\(D = \R^2 \times \R\)
the foliation is the map \(\bigcup_{\R} \R \rightarrow \R^2\)