A Riemannian manifold is a smooth manifold equipped with a Riemannian metric, which is a smoothly varying choice of inner product on each tangent space. Essentially, it's a space that locally resembles Euclidean space but may have curvature. This concept, conceptualized by Bernhard Riemann in 1854, allows for the use of differential and integral calculus to extract geometric data.
The Riemannian metric provides a way to measure lengths and angles within the manifold, defining the shortest distance (geodesics) between points. Examples of Riemannian manifolds include spheres, hyperbolic spaces, and smooth surfaces in three-dimensional space. Riemannian manifolds have applications in various fields, including engineering, physics, and machine learning, particularly in areas like global analysis, differential equations, and manifold learning. They serve as a natural extension of Euclidean space, providing a framework for studying curved spaces and their properties.
