Gauss proved curvature can be felt from inside a surface
A Riemannian manifold carries smoothly varying inner products on tangent spaces so distance, angle, length, volume, and curvature make sense. Euclidean space, spheres, hyperbolic space, and smooth surfaces such as ellipsoids are examples. Bernhard Riemann introduced the idea non-rigorously in 1854; the name follows him.
Gauss found in 1827 that a surface’s curvature can be computed from measurements taken entirely within it, the first fundamental form, a result he called the Theorema Egregium, or remarkable theorem. Maps that preserve those local measurements are local isometries; properties they keep are intrinsic and those they can destroy are extrinsic, so curvature belongs to the surface itself rather than to how it bends through space.
Riemann sketched such spaces and their curvature informally in 1854, but rigour arrived slowly: Hermann Weyl’s 1913 book was the first to spell out even the more basic idea of a smooth manifold. Élie Cartan introduced one of the earliest connections, and Tullio Levi-Civita defined the special connection that bears his name. Einstein built general relativity on the pseudo-Riemannian generalisation, with his field equations constraining the curvature of four-dimensional spacetime.
A bare smooth manifold has no inner product on its tangent vectors, and without one calculus cannot measure the length of a curve. A Riemannian metric supplies that measuring stick at every point as a smoothly varying positive-definite symmetric bilinear form, a special kind of metric tensor, quite different from the distance function that topologists also call a metric. Integrating it yields distances between points, while differentiating it produces curvature. In coordinates it becomes a symmetric positive-definite matrix at each point, and analysts also study rougher Lipschitz or merely measurable versions.
Surfaces in ordinary space, and submanifolds of any dimension, inherit metrics from the surrounding dot product, and John Nash proved every Riemannian manifold can be realised that way. The subject links to topology, complex and algebraic geometry and reaches gauge theory, machine learning and cartography, with generalisations called Finsler and sub-Riemannian manifolds.
Source: Riemannian manifold
