= Symplectic splitting along a Lagrangian submanifold
{title2=$TM|_X\cong TX\oplus T^*X$}
A <Lagrangian complement> to $TX$ in $TM|_X$ is identified with $T^*X$ by $v\mapsto\omega(\,\cdot\,,v)|_{TX}$. This produces a <vector bundle isomorphism> $TX\oplus T^*X\to TM|_X$ 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>.
Back to article page
