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Manifolds: a global setting for geometry and analysis, with or without coordinates
Author(s) -
Goldman William M.
Publication year - 2012
Publication title -
wiley interdisciplinary reviews: computational statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.693
H-Index - 38
eISSN - 1939-0068
pISSN - 1939-5108
DOI - 10.1002/wics.1220
Subject(s) - non euclidean geometry , riemannian geometry , euclidean geometry , geometry , mathematics , euclidean space , foundations of geometry , generalization , absolute geometry , dimension (graph theory) , ordered geometry , context (archaeology) , analytic geometry , synthetic geometry , pure mathematics , algebra over a field , differential geometry , mathematical analysis , projective geometry , geography , convex set , archaeology , convex optimization , regular polygon
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