The Riemannian metric

A Riemannian metric is a mathematical tool used to measure distances and angles on curved surfaces or spaces. Think of it as a way to define what “straight,” “short,” or “close” means on a shape that isn’t flat, like the surface of a globe. Instead of relying on regular rulers, the Riemannian metric adapts to the curvature, allowing precise measurement of lengths and angles, much like a flexible measuring system that respects the shape’s geometry. This concept helps mathematicians understand and analyze the intrinsic properties of curved spaces in a rigorous way.