Isometries

Differential Geometry · Axiom Academy

LESSON Isometries in Differential Geometry Understanding distance-preserving maps between surfaces and their fundamental properties 1. Definition: Distance-Preserving Maps A map f: S 1 → S 2 between surfaces is called an isometry if it preserves the length of all curves. Equivalently, for any two points p, q ∈ S 1 , the distance between them equals the distance between their images. This means the metric structure is completely preserved under the transformation. 2. Isometry Preserves the First Fundamental Form A map is an isometry if and only if it preserves the first fundamental form. This means that in local coordinates, the metric coefficients are identical on both surfaces. Equivalently, the coefficients of the first fundamental form satisfy: This is the practical criterion for checking whether two surfaces are locally isometric: their first fundamental forms must match under appropriate parameterizations. 3. Global Isometries Between Surfaces A global isometry is a bijective isometry between entire surfaces. Two surfaces are globally isometric if there exists a global isometry between them. Key Property: Globally isometric surfaces have identical intrinsic geometries. This includes: Gaussian curvature at corresponding points Geodesics (intrinsic "straight lines") All intrinsic measurements (angles, areas, distances) Important: Surfaces can be extrinsically different (different shapes in 3D) but intrinsically identical (same measurements on the surface).

This is the written version of the interactive lesson above. See the full Differential Geometry course.