Ambient space (mathematics) explained

(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 ( R ^{2)} —in which case the ambient space of l is R ^{2} , or as an object embedded in 2-dimensional hyperbolic space ( H ^{2)} —in which case the ambient space of l is 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} , but false if the ambient space is H ^{2} , because the geometric properties of R ^{2} are different from the geometric properties of 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

• Book: Schilders, W. H. A. . E. J. W. . ter Maten . Philippe G. . Ciarlet . Numerical Methods in Electromagnetics . Special Volume . . 2005 . 0-444-51375-2 . 120ff .
• Book: Wiggins, Stephen . Stephen Wiggins . Chaotic Transport in Dynamical Systems . registration . Berlin . Springer . 1992 . 3-540-97522-5 . 209ff .