STARLIGHT
THEORY

Riemannian Metrics and Distance

Riemannian Metrics: The Art of Measurement

How do we measure distances on a curved surface? How do we define angles between curves? Riemannian metrics provide the framework for answering these questions and many more.

Understanding Riemannian Metrics

A Riemannian metric gives us a way to measure lengths and angles at each point of a manifold. It’s like equipping each tangent space with its own inner product, but in a smooth way.

What We’ll Explore:

  1. Definition of a Riemannian metric
  2. Length of curves
  3. Geodesics
  4. The Levi-Civita connection
  5. Curvature

Applications include:

  • General relativity
  • Image processing
  • Computer graphics
  • Machine learning

This post sets the foundation for understanding geometry in curved spaces.