Riemann made geometry’s measuring stick vary point by point
Riemannian geometry studies Riemannian manifolds—spaces where distance is length of curves on the space itself, generalising curved surfaces. Bernhard Riemann sketched the vision in his inaugural lecture On the Hypotheses on which Geometry is Based. Metrics can change from point to point, embracing non-Euclidean worlds.
The metric starts from a familiar idea: restrict the ordinary dot product of surrounding space to vectors tangent to a surface. Riemann’s leap was to insist that this quadratic form, not the surface’s particular shape in space, is what counts, so a rolled cylinder and a flat sheet share the same intrinsic geometry, and a manifold need not sit inside any Euclidean space at all. Every smooth manifold can carry many such metrics, and their properties restrict its topology. Swapping the quadratic form for a more general function leads to Finsler geometry, while letting some non-zero vectors have zero length gives the pseudo-Riemannian spaces of general relativity.
Two landmark results connect the abstract and the concrete. The Gauss–Bonnet theorem says the total Gaussian curvature of a compact surface equals 2π times its Euler characteristic, a statement later generalised to all compact even-dimensional manifolds. The Nash embedding theorems show that any Riemannian manifold can nonetheless be placed isometrically inside some Euclidean space.
Many classic theorems infer global shape from local curvature. The sphere theorem says a simply connected compact manifold whose sectional curvature is strictly pinched between one quarter and one must be a sphere. Cartan–Hadamard shows that a complete simply connected space with nonpositive curvature is just Euclidean space in disguise, with a unique geodesic between any two points. Myers’ theorem makes the fundamental group finite when Ricci curvature is positive, and on compact spaces of negative curvature the geodesic flow is ergodic.
The Cheeger–Gromoll soul theorem finds a compact core inside every complete, non-compact, non-negatively curved manifold, and Grigori Perelman gave a strikingly short proof of the related soul conjecture in 1994. Cheeger’s finiteness theorem limits how many manifolds fit given bounds on curvature, diameter and volume.
Source: Riemannian geometry
