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Motion (geometry)
In geometry, a motion is an isometry of a metric space. For instance, a plane with Euclidean distance as metric is a metric space in which a mapping associating congruent figures is a motion.〔Gunter Ewald (1971) ''Geometry: An Introduction'', p. 179, Belmont: Wadsworth ISBN 0-534-00034-7〕 More generally, the term ''motion'' is a synonym for surjective isometry in metric geometry,〔M.A. Khamsi & W.A. Kirk (2001) ''An Introduction to Metric Spaces and Fixed Point Theorems'', p. 15, John Wiley & Sons ISBN 0-471-41825-0〕 including elliptic geometry and hyperbolic geometry. In the latter case, hyperbolic motions provide an approach to the subject for beginners.
==In differential geometry==
In differential geometry, a diffeomorphism is called a motion if it induces an isometry between the tangent space at a manifold point and the tangent space at the image of that point.〔A.Z. Petrov (1969) ''Einstein Spaces'', p. 60, Pergamon Press〕〔B.A. Dubrovin, A.T. Fomenko, S.P Novikov (1992) ''Modern Geometry – Methods and Applications'', second edition, p 24, Springer, ISBN 978-0-387-97663-1〕
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