garywelz/programming_framework

1<!DOCTYPE html>2<html lang="en">3<head>4    <meta charset="UTF-8">5    <meta name="viewport" content="width=device-width, initial-scale=1.0">6    <title>Connections on Manifolds - Mathematics Process</title>7    <script src="https://cdn.jsdelivr.net/npm/mermaid@10.6.1/dist/mermaid.min.js"></script>8    <style>9        * { margin: 0; padding: 0; box-sizing: border-box; }10        body { font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif; background: linear-gradient(135deg, #667eea 0%, #764ba2 100%); min-height: 100vh; padding: 20px; }11        .container { max-width: 1200px; margin: 0 auto; background: white; border-radius: 15px; box-shadow: 0 20px 40px rgba(0,0,0,0.1); overflow: hidden; }12        .header { background: #d35400; color: white; padding: 30px; }13        .header h1 { margin: 0 0 10px 0; font-size: 2em; font-weight: 300; }14        .header-meta { display: flex; flex-wrap: wrap; gap: 15px; margin-top: 15px; font-size: 0.9em; opacity: 0.9; }15        .meta-item { background: rgba(255,255,255,0.2); padding: 5px 12px; border-radius: 20px; }16        .nav-links { padding: 15px 30px; background: #f8f9fa; border-bottom: 1px solid #ecf0f1; }17        .nav-links a { color: #d35400; text-decoration: none; margin-right: 20px; font-weight: 500; }18        .nav-links a:hover { text-decoration: underline; }19        .content { padding: 30px; }20        .description { margin-bottom: 30px; }21        .flowchart-container { margin: 30px 0; }22        .flowchart-container h2 { color: #2c3e50; margin-bottom: 15px; }23        .mermaid { background: white; padding: 20px; border-radius: 10px; border: 1px solid #ecf0f1; overflow-x: auto; }24        .color-legend { background: #f8f9fa; padding: 20px; border-radius: 10px; margin: 30px 0; }25        .color-legend h3 { color: #2c3e50; margin-bottom: 15px; }26        .color-grid { display: grid; grid-template-columns: repeat(auto-fit, minmax(180px, 1fr)); gap: 15px; }27        .color-item { display: flex; align-items: center; gap: 10px; padding: 10px; background: white; border-radius: 5px; }28        .color-box { width: 30px; height: 30px; border-radius: 4px; border: 1px solid #ddd; }29        .info-card { background: #f8f9fa; padding: 20px; border-radius: 10px; margin-top: 20px; }30        .info-card h3 { color: #2c3e50; margin-bottom: 15px; }31        .info-card ul { list-style: none; padding: 0; }32        .info-card li { padding: 8px 0; border-bottom: 1px solid #ecf0f1; }33        .info-card li:last-child { border-bottom: none; }34    </style>35</head>36<body>37    <div class="container">38        <div class="header">39            <h1>Connections on Manifolds</h1>40            <div class="header-meta">41                <span class="meta-item">Differential Geometry</span>42                <span class="meta-item">Mathematics</span>43            </div>44        </div>45        <div class="nav-links">46            <a href="../../mathematics_index.html">← Back to Mathematics Index</a>47            <a href="differential_geometry-index.html">Differential Geometry Index</a>48            <a href="https://arxiv.org/list/math.DG/recent" target="_blank">arXiv: Differential Geometry (math.DG)</a>49        </div>50        <div class="content">51            <div class="description">52                <h2>Description</h2>53                <p>Affine connections, parallel transport, holonomy, and curvature. Dependency graph for covariant derivatives and the structure equations.</p>54            </div>55            <div class="flowchart-container">56                <h2>Dependency Flowchart</h2>57                <div class="mermaid">58graph TD59    D1["D1 Affine connection ∇\nKoszul axioms: C^∞-linear in X, Leibniz in Y"]60    D2["D2 Parallel transport\nP_γ: T_p M → T_q M along γ"]61    D3["D3 Holonomy group Hol_p\nCurvature at p from parallel transport"]62    D4["D4 Torsion tensor T\nT(X,Y) = ∇_X Y − ∇_Y X − [X,Y]"]63    64    T1["T1 Existence of parallel sections\nFlat connection ⇒ local parallel frames"]65    T2["T2 Ambrose-Singer\nHolonomy algebra = curvature derivatives"]66    T3["T3 Frobenius for distributions\nInvolutive ⇒ integrable"]67    T4["T4 Ricci identity\n[∇_X, ∇_Y]σ − ∇_[X,Y] σ = R(X,Y)σ"]68    T5["T5 Geodesic equation\nConnection defines autoparallel curves"]69    70    D1 --> D271    D1 --> D472    D2 --> D373    D1 --> T174    D3 --> T275    D1 --> T376    D1 --> T477    D1 --> T578    D2 --> T179    D4 --> T480    81    classDef definition fill:#3498db,color:#fff,stroke:#2980b982    classDef theorem fill:#1abc9c,color:#fff,stroke:#16a08583    class D1,D2,D3,D4 definition84    class T1,T2,T3,T4,T5 theorem85                </div>86            </div>87            <div class="color-legend">88                <h3>Color Scheme</h3>89                <div class="color-grid">90                    <div class="color-item"><div class="color-box" style="background:#3498db"></div><div><strong>Blue</strong><br><small>Definitions (D1–D4)</small></div></div>91                    <div class="color-item"><div class="color-box" style="background:#1abc9c"></div><div><strong>Teal</strong><br><small>Theorems (T1–T5)</small></div></div>92                </div>93            </div>94            <div class="info-card">95                <h3>Info</h3>96                <ul>97                    <li><strong>Subcategory:</strong> differential_geometry</li>98                    <li><strong>Keywords:</strong> connection, parallel transport, holonomy, torsion, curvature</li>99                    <li><strong>Research frontier:</strong> <a href="https://arxiv.org/list/math.DG/recent" target="_blank">arXiv math.DG</a></li>100                </ul>101            </div>102        </div>103    </div>104    <script>105        mermaid.initialize({ startOnLoad: true, theme: 'default', flowchart: { useMaxWidth: true, htmlLabels: true, curve: 'step', nodeSpacing: 25, rankSpacing: 40 } });106    </script>107</body>108</html>109