Intrinsic vs Extrinsic

Differential Geometry · Axiom Academy

INTRO Intrinsic vs Extrinsic Geometry Discover how surfaces have properties that depend on perspective: from within or from outside. Imagine rolling a flat piece of paper into a cylinder. Adjust the radius and see what happens to the surface. Imagine you're a 2D being living on a surface. You can only measure things from within—no looking outside! Step 3: Gaussian Curvature is Intrinsic Here's a remarkable fact: Gaussian curvature can be determined entirely from within the surface! If you bend a surface without stretching or tearing it (an isometric deformation ), the Gaussian curvature remains unchanged. This is because Gaussian curvature depends only on the metric—the way distances are measured on the surface. Step 4: Mean Curvature is Extrinsic Unlike Gaussian curvature, mean curvature depends on how the surface sits in 3D space. Measurable from within the surface using only the metric (distance measurements). Independent of how the surface sits in space. Depend on the embedding of the surface in ambient space. Require looking from outside. What appears to be extrinsic (curvature) can sometimes be intrinsic! The Theorema Egregium revolutionized our understanding of geometry by showing that Gaussian curvature is a property of the surface itself, not its embedding.

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