Riemannian Geometry

Riemannian geometry equips a smooth manifold with a metric that measures lengths and angles, turning the calculus of manifolds into a genuine geometry of distance, geodesics, and curvature.

Definition

Riemannian geometry is the study of smooth manifolds equipped with a Riemannian metric — a smoothly varying inner product on tangent spaces — and the geometric notions of length, angle, geodesic, and curvature that the metric determines.

Scope

This area covers manifolds endowed with a Riemannian metric: the Levi-Civita connection and parallel transport, geodesics as locally shortest paths, the curvature tensor and its contractions (sectional, Ricci, and scalar curvature), and the global comparison theorems relating curvature bounds to topology and to distance. It includes the interplay between local curvature and global shape that motivates much of modern geometry, while excluding the metric-free smooth structures of differential topology and the indefinite metrics studied in Lorentzian geometry.

Sub-topics

• Connections and Parallel Transport
• Curvature and Comparison Geometry
• Riemannian Metrics and Geodesics

Core questions

• How does a metric determine a unique compatible, torsion-free connection (Levi-Civita) and hence geodesics?
• What are the different curvatures, and how do they encode the local deviation from flatness?
• How do curvature bounds constrain the global topology and diameter of a manifold?
• When are two Riemannian manifolds isometric, and which quantities are isometry invariants?

Key concepts

• Riemannian metric and isometries
• Levi-Civita connection and parallel transport
• Geodesics and the exponential map
• Riemann curvature tensor, sectional, Ricci, and scalar curvature
• Comparison theorems relating curvature to topology

Clinical relevance

Riemannian geometry is the mathematical framework of general relativity (with its Lorentzian generalization), underlies geometric analysis and Ricci-flow techniques used to resolve the Poincaré conjecture, and provides the curved metrics central to optimization, shape analysis, and machine learning on manifolds.

History

Riemann's 1854 habilitation lecture introduced the metric notion of curvature in arbitrary dimensions; Levi-Civita's parallel transport (1917) gave the connection its geometric meaning, and the global comparison geometry developed by Cartan, Rauch, and later Gromov turned the subject into the study of curvature versus topology.

Key figures

• Bernhard Riemann
• Tullio Levi-Civita
• Mikhail Gromov

Seminal works

• lee1997
• docarmo1992

Frequently asked questions

What does a Riemannian metric add to a smooth manifold?

It supplies an inner product on each tangent space, varying smoothly, which lets one measure lengths of curves, angles between vectors, volumes, and ultimately curvature — none of which exist on a bare smooth manifold.

How is Riemannian geometry related to general relativity?

General relativity uses a pseudo-Riemannian (Lorentzian) metric of indefinite signature on spacetime; the Levi-Civita connection, geodesics, and curvature tensor of Riemannian geometry carry over and describe free fall and gravitation as curvature.