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:
- Definition of a Riemannian metric
- Length of curves
- Geodesics
- The Levi-Civita connection
- Curvature
Applications include:
- General relativity
- Image processing
- Computer graphics
- Machine learning
This post sets the foundation for understanding geometry in curved spaces.