Ambient space

An ambient space or ambient configuration space is the space surrounding an object.

Mathematics

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

In mathematics, especially in geometry and topology, an ambient space is the space surrounding a mathematical object. For example, a line may be studied in isolation, or it may be studied as an object in two-dimensional space — in which case the ambient space is the plane, or as an object in three-dimensional space — in which case the ambient space is three-dimensional. To see why this makes a difference, consider the statement "Lines that never meet are necessarily parallel." This is true if the ambient space is two-dimensional, but false if the ambient space is three-dimensional, because in the latter case the lines could be skew lines, rather than parallel.

See also

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

Further reading

• W H A Schilders, E.J .W. ter Maten, Philippe G. Ciarlet, Numerical Methods in Electromagnetics: Special Volume, Elsevier 2005. (ed., with particular attention to page 120+.)
• Stephen Wiggins, Chaotic Transport in Dynamical Systems. 1992. (ed., with particular attention to page 209+.)
• "Relative Hyperbolicity, Trees of Spaces and Cannon–Thurston Maps" arXiv:0708.3578, 2007
• "Relative Hyperbolic Extensions of Groups and Cannon–Thurston Maps" arXiv:0801.0933, 2008