What is Intrinsic Geometry?

Differential Geometry · Axiom Academy

INTRO What is Intrinsic Geometry? Discover how to measure a curved world from the inside. Imagine you're an ant living on a sphere. You can't step off the surface to see it from the outside. Drag the point to explore how the ant experiences the sphere. The ant measures distance by walking along the surface. Move the slider to trace a path between two points on the sphere. Like distance, angles between curves and areas of regions are intrinsic. Adjust the slider to see how a small triangle on the sphere has measurable angles and area. The ant's "straight lines" are called geodesics - the shortest paths between points on the surface. Click "Trace Geodesic" to see one. If the ant carries a compass while walking, how does the direction change? Click "Start Transport" to see parallel transport in action. Intrinsic geometry studies properties measurable from within a surface - distance, angle, area, geodesics, curvature. Extrinsic geometry requires an outside perspective - like how the surface bends in 3D space. Distance along curves, angles between tangent vectors, and areas of regions are all intrinsic. They can be measured by inhabitants who never leave the surface. You'll learn about geodesics (intrinsic "straight lines"), parallel transport (how vectors change along curves), and the Riemann curvature tensor - the ultimate intrinsic measure of how space curves.

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