Ambient space (mathematics)

"Ambient space" redirects here. Not to be confused with Ambient space (psychology).

Three examples of different geometries: Euclidean, elliptical, and hyperbolic

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 $(l)$ may be studied in isolation —in which case the ambient space of $l$ is $l$, or it may be studied as an object embedded in 2-dimensional Euclidean space $(\mathbb {R} ^{2})$—in which case the ambient space of $l$ is $\mathbb {R} ^{2}$, or as an object embedded in 2-dimensional hyperbolic space $(\mathbb {H} ^{2})$—in which case the ambient space of $l$ is $\mathbb {H} ^{2}$. To see why this makes a difference, consider the statement "Parallel lines never intersect." This is true if the ambient space is $\mathbb {R} ^{2}$, but false if the ambient space is $\mathbb {H} ^{2}$, because the geometric properties of $\mathbb {R} ^{2}$ are different from the geometric properties of $\mathbb {H} ^{2}$. 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). Chaotic Transport in Dynamical Systems. Berlin: Springer. pp. 209ff. ISBN 3-540-97522-5.