Ambient space (mathematics)

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 the real line, 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

• Configuration space
• Geometric space
• Manifold and ambient manifold
• Submanifolds and Hypersurfaces
• Riemannian manifolds
• Ricci curvature
• Differential form

Further reading

• Schilders, 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.
• Wiggins, Stephen (1992). . Berlin: Springer. pp. 209ff. ISBN 3-540-97522-5.