Riemannian Manifolds

A Riemannian manifold is a mathematical space that generalizes familiar shapes like curves and surfaces to higher dimensions, equipped with a way to measure distances and angles smoothly at every point. Think of it as a landscape with a consistent notion of "straightness" and "closeness," even if the surface is curved or complex. This structure allows mathematicians and physicists to analyze and understand the geometry of spaces that are not flat, enabling tools like measuring shortest paths (geodesics), curvature, and volume, which are essential in areas like relativity, robotics, and advanced geometry.