Riemannian Metrics

A Riemannian metric is a mathematical way to measure distances and angles on curved surfaces or spaces, extending the idea of geometry beyond flat surfaces. Think of it as a tool that allows us to define how to "walk" on a curved plane, like the surface of a sphere. It provides a way to assign a length to paths, helping us understand concepts like curvature. In the context of general relativity, Riemannian metrics help describe the geometry of space-time, revealing how mass and energy influence the shape of the universe.