Manifolds: a global setting for geometry and analysis, with or without coordinates

Manifolds are curved spaces that admit local coordinate systems which look like Euclidean space. Much of the classical geometry of Euclidean space extends and generalizes to manifolds. A generalization of Euclidean geometry is Riemannian geometry, where distances are defined which, in small regions, look like Euclidean geometry. This article describes how familiar concepts of Euclidean geometry generalize to the context of Riemannian manifolds, as well as more general structures. WIREs Comput Stat 2012 doi: 10.1002/wics.1220 This article is categorized under: Statistical and Graphical Methods of Data Analysis > Dimension Reduction