Charts and Atlases

Differential Geometry · Axiom Academy

Understanding coordinate systems on manifolds through charts, atlases, and transition functions 1. Charts (Coordinate Systems) A chart is a coordinate system that maps a region of a manifold to Euclidean space. U is an open subset of M (the coordinate neighborhood) φ: U → ℝⁿ is a homeomorphism onto its image The map φ assigns coordinates to each point in U, allowing us to do calculations in familiar Euclidean space. 2. Atlases: Collections of Compatible Charts A single chart typically cannot cover an entire manifold. We need a collection of charts that work together. The coordinate neighborhoods cover M: ⋃_i U_i = M On overlaps U_i ∩ U_j, the charts are "compatible" (smoothly related) The coverage requirement ensures every point on the manifold has coordinates from at least one chart. When two charts overlap, we need to relate their coordinate systems. This relationship is captured by transition functions . φ_j ∘ φ_i⁻¹: φ_i(U_i ∩ U_j) → φ_j(U_i ∩ U_j) Maps between coordinate representations in ℝⁿ For a smooth manifold, all transition functions must be smooth (infinitely differentiable). 4. Example: Stereographic Projection on the Sphere The 2-sphere S² provides a classic example demonstrating why we need multiple charts. Projects sphere onto equatorial plane ℝ² Draw line from north pole through point p φ_N(p) = intersection with plane Similarly, we can project from the south pole to get φ_S. Together, (U_N, φ_N), (U_S, φ_S) form an atlas for S².

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