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In mathematics, especially in geometry and topology, an ambient space is the space surrounding a mathematical object along with the object itself. For example, a 1-dimensional line ( l ) {\displaystyle (l)} may be studied in isolation —in which case the ambient space of l {\displaystyle l} is l {\displaystyle l} , or it may be studied as an object embedded in 2-dimensional Euclidean space ( R 2 ) {\displaystyle (\mathbb {R} ^{2})} —in which case the ambient space of l {\displaystyle l} is R 2 {\displaystyle \mathbb {R} ^{2}} , or as an object embedded in 2-dimensional hyperbolic space ( H 2 ) {\displaystyle (\mathbb {H} ^{2})} —in which case the ambient space of l {\displaystyle l} is H 2 {\displaystyle \mathbb {H} ^{2}} . To see why this makes a difference, consider the statement "Parallel lines never intersect." This is true if the ambient space is R 2 {\displaystyle \mathbb {R} ^{2}} , but false if the ambient space is H 2 {\displaystyle \mathbb {H} ^{2}} , because the geometric properties of R 2 {\displaystyle \mathbb {R} ^{2}} are different from the geometric properties of H 2 {\displaystyle \mathbb {H} ^{2}} . All spaces are subsets of their ambient space.

• See also

• Further reading

• Configuration space

• Geometric space

• Manifold and ambient manifold

• Submanifolds and Hypersurfaces

• Riemannian manifolds

• Ricci curvature

• Differential form

• citerefschilderster-matenciarlet2005Schilders, W. H. A.; ter Maten, E. J. W.; Ciarlet, Philippe G. (2005). Numerical Methods in Electromagnetics. Vol. Special Volume. Elsevier. pp. 120ff. ISBN 0-444-51375-2.

• citerefwiggins1992Wiggins, Stephen (1992). Chaotic Transport in Dynamical Systems. Berlin: Springer. pp. 209ff. ISBN 3-540-97522-5.