Riemannian manifold

A Riemannian manifold is a smooth manifold equipped with a positive-definite inner product on each tangent space that varies smoothly from point to point — the metric tensor. It is the foundational object of differential geometry and the setting in which Ricci curvature, sectional curvature, and scalar curvature are defined. Unlike the more general spaces studied in metric geometry, a Riemannian manifold assumes infinite differentiability, but this smoothness is increasingly understood as a convenience rather than a necessity: the deep theorems of comparison geometry depend only on curvature bounds, not on derivatives. The metric determines a unique torsion-free, metric-compatible connection that defines parallel transport and geodesics, making Riemannian manifolds self-contained geometric worlds that need no ambient space.