Local Isometries

Differential Geometry · Axiom Academy

Understanding distance-preserving maps and their role in differential geometry 1. Definition: Locally Distance-Preserving A smooth map f: S → S' between surfaces is a local isometry if for every point p ∈ S, there exists a neighborhood U of p such that f restricted to U preserves arc lengths of curves. Formally: For any curve γ(t) in U, the length of γ equals the length of f(γ). This means the first fundamental forms are equal: I = I'. 2. Difference from Global Isometry A global isometry is a bijective local isometry - it's distance-preserving everywhere AND one-to-one. A local isometry may fail to be injective or surjective. Key Example: Wrapping a plane around a cylinder. Locally distances are preserved, but the map is not injective (many points on the plane map to the same point on the cylinder). 3. Example: Helicoid to Catenoid The helicoid and catenoid are locally isometric surfaces - they have the same Gaussian curvature at corresponding points, so small patches can be mapped between them preserving distances. However, they are NOT globally isometric because they have different topological properties (the helicoid is not closed, while a full catenoid wraps around). 4. Developable Surfaces (K = 0) A surface with zero Gaussian curvature (K = 0) is called developable . These surfaces are locally isometric to the plane - they can be "unrolled" flat without distortion.

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