Intrinsic Geometry Summary

Differential Geometry · Axiom Academy

SUMMARY Intrinsic Geometry Summary Let's review the fundamental concepts that define intrinsic geometry on curved surfaces. Definition: Intrinsic properties can be measured from within the surface, without reference to any ambient space 2D Perspective: Imagine being a 2D creature confined to the surface—intrinsic properties are what you can measure Key Examples: Distances along curves, angles between curves, area, and Gaussian curvature Not Intrinsic: Normal vectors, mean curvature, and the embedding itself depend on the ambient 3D space Definition: Smooth maps between surfaces that preserve distances (and therefore all intrinsic properties) What They Preserve: Arc lengths, angles, areas, geodesics, and Gaussian curvature Classic Example: Rolling a sheet of paper into a cylinder—distances and angles remain unchanged Why It Matters: Surfaces related by isometries are intrinsically identical, even if they look different in 3D space Role: Encode how basis vectors change as you move across the surface Formula: Γ k ij measures the rate of change of tangent vectors in the coordinate basis Computed From: First and second derivatives of the metric tensor (the first fundamental form) Enables: The covariant derivative, which lets us differentiate vector fields intrinsically Geodesics as Straightest Paths Definition: Curves that parallel-transport their own tangent vector Equivalent View: Locally shortest paths between points on the surface

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