A **symplectic manifold** is a smooth manifold \( M \) equipped with a closed non-degenerate differential 2-form called the **symplectic form**, typically denoted by \( \omega \). Formally, a symplectic manifold is defined as follows: 1. **Manifold**: \( M \) is a differentiable manifold of even dimension, usually denoted as \( 2n \), where \( n \) is a positive integer.
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A symplectic manifold is an even-dimensional smooth manifold equipped with a closed nondegenerate differential two-form .