Symplectic Manifolds

This post was translated from Korean by LLM (Kimi). The translation may contain errors or awkward sentences. The Korean original is the source of truth.

In [MS], symplectic vector spaces are introduced, after which more time is spent on the Maslov class, among other topics. We will introduce these later when they are needed for Floer theory, and for now follow [Cd] to first define symplectic manifolds.

Definition 1 A symplectic form \(\omega\) on a manifold \(M\) is a differential \(2\)-form such that \(d\omega=0\) and, for every \(p\in M\), \(\omega_p:T_pM\times T_pM\rightarrow \mathbb{R}\) is a linear symplectic form. We then call \((M,\omega)\) a symplectic manifold.

For a symplectic manifold \((M,\omega)\), the vector space \(T_pM\) is equipped with a linear symplectic form \(\omega_p\), so \(\dim T_pM\) is even, and therefore \(M\) must also be even-dimensional.