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 the real line, 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 (in particular, curvature) 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
• Abstract variety and Projective embedding

Further reading

• Schilders, W. H. A.; ter Maten, E. J. W.; Ciarlet, Philippe G., eds. (2005). Numerical Methods in Electromagnetics. Handbook of Numerical Analysis. Vol. 13. Amsterdam: North Holland. ISBN 978-0-444-51375-5.

• Wiggins, Stephen (1992). Chaotic Transport in Dynamical Systems. Interdisciplinary Applied Mathematics. Vol. 2. New York: Springer. ISBN 978-0-387-97522-1.