What Is a Manifold? | Quanta Magazine

Mathematicians call certain shapes 'manifolds' – spaces that appear flat when viewed locally but can possess complex global structures. Introduced by Bernhard Riemann in the mid-19th century, manifolds revolutionized the study of mathematical space, leading to modern topology and finding applications in geometry, dynamical systems, data analysis, and physics, including Einstein's theory of relativity. The article explains how manifolds allow mathematicians to apply traditional calculus techniques to curved or higher-dimensional spaces by dividing them into 'charts' that map to Euclidean space, offering a fundamental vocabulary for understanding and solving diverse problems across scientific disciplines.