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| <h1>Symplectic Form</h1> | |
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| <span class="meta-item">Symplectic Geometry</span> | |
| <span class="meta-item">Mathematics</span> | |
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| <a href="../../mathematics_index.html">← Back to Mathematics Index</a> | |
| <a href="symplectic_geometry-index.html">Symplectic Geometry Index</a> | |
| <a href="https://arxiv.org/list/math.SG/recent" target="_blank">arXiv: Symplectic Geometry (math.SG)</a> | |
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| <h2>Description</h2> | |
| <p>Symplectic manifold (M, ω): nondegenerate closed 2-form. Dimension even, ω^n volume form. Linear symplectic algebra and Darboux basis.</p> | |
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| <h2>Dependency Flowchart</h2> | |
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| graph TD | |
| D1["D1 Symplectic form ω\nClosed, nondegenerate 2-form"] | |
| D2["D2 Nondegeneracy\nω^n ≠ 0; ω^∧: TM → T*M iso"] | |
| D3["D3 Symplectic vector space\n(V, ω) with dim V even"] | |
| D4["D4 Symplectomorphism\nφ*ω = ω, preserves form"] | |
| T1["T1 Dimension even\ndim M = 2n for symplectic M"] | |
| T2["T2 ω^n is volume form\nOriented manifold"] | |
| T3["T3 Linear Darboux\n(V,ω) ≅ (R^{2n}, ω_0) standard"] | |
| T4["T4 Darboux theorem\nLocally (M,ω) ≅ (R^{2n}, ω_0)"] | |
| T5["T5 Symplectic quotient\nM // G inherits symplectic structure"] | |
| D1 --> D2 | |
| D1 --> D3 | |
| D1 --> D4 | |
| D2 --> T1 | |
| D2 --> T2 | |
| D3 --> T3 | |
| D1 --> T4 | |
| D4 --> T5 | |
| T3 --> T4 | |
| T1 --> T2 | |
| classDef definition fill:#3498db,color:#fff,stroke:#2980b9 | |
| classDef theorem fill:#1abc9c,color:#fff,stroke:#16a085 | |
| class D1,D2,D3,D4 definition | |
| class T1,T2,T3,T4,T5 theorem | |
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| <h3>Color Scheme</h3> | |
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| <div class="color-item"><div class="color-box" style="background:#3498db"></div><div><strong>Blue</strong><br><small>Definitions (D1–D4)</small></div></div> | |
| <div class="color-item"><div class="color-box" style="background:#1abc9c"></div><div><strong>Teal</strong><br><small>Theorems (T1–T5)</small></div></div> | |
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| <h3>Info</h3> | |
| <ul> | |
| <li><strong>Subcategory:</strong> symplectic_geometry</li> | |
| <li><strong>Keywords:</strong> symplectic form, Darboux, nondegenerate, symplectomorphism</li> | |
| <li><strong>Research frontier:</strong> <a href="https://arxiv.org/list/math.SG/recent" target="_blank">arXiv math.SG</a></li> | |
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