Weinstein neighborhood theorem

A neighborhood of a compact Lagrangian submanifold $L\subset (M,\omega )$ is symplectomorphic, by a map restricting to the identity on $L$, to a neighborhood of the zero section in $(T^{\ast }L,d\lambda )$, with the sign chosen to match the convention for the canonical form.

A Lagrangian complement to $TX$ in $TM{|}_X$ is identified with $T^{\ast }X$ by $v\mapsto \omega (\cdot ,v){|}_{TX}$. This produces a vector bundle isomorphism $TX\oplus T^{\ast }X\to TM{|}_X$ that is the identity on $TX$ and carries the canonical pairing $\zeta (u)-\eta (v)$ to $\omega$. A suitably chosen tubular neighborhood map therefore matches the ambient two-forms along $X$, enabling the relative Moser theorem.

If a compact Lagrangian $L$ has $H^1(L;\mathbb{R})=0$, every sufficiently close Lagrangian graph is the graph of $df$ and meets the zero section at least at a maximum and a minimum of $f$. Without the cohomology assumption, a nowhere-zero closed one-form can give a disjoint nearby graph.

A compact Lagrangian submanifold $L$ with vanishing first de Rham cohomology cannot be a fibre of a smooth positive-dimensional fiber bundle whose nearby fibres are also Lagrangian submanifolds. In a Weinstein neighborhood, nearby fibres are one-form graphs of closed forms. They are exact by the cohomology assumption, so meet the zero section at a critical point of their potential, contradicting disjointness of distinct fibres.