Introduction
Riemannian geometry is the branch of differential geometry that studies smooth manifolds equipped with a Riemannian metric—an inner product on each tangent space that varies smoothly from point to point, thereby endowing the manifold with notions of angle, length of curves, surface area and volume. Originating in Bernhard Riemann’s 1854 lecture “On the Hypotheses which underlie Geometry,” it generalizes the classical differential geometry of curves and surfaces to spaces of arbitrary dimension, introducing curvature as a fundamental invariant that measures local deviation from flatness. This framework not only underpins Einstein’s formulation of gravitation in general relativity but also provides powerful tools in geometric analysis, global topology and the study of geodesics on curved spaces.
Contents
Riemannian Metrics
Geodesics and Completeness
Torsion and Curvature
The Levi-Civita Connection
Riemannian Geodesics and Completeness
Warped Products
Conformal