View source for Ambient space (mathematics)

{{short description|The space surrounding an object}}
{{distinguish-redirect|Ambient space|Ambient space (psychology)}}
{{inline |date=May 2024}}
[[File:Euclidian and non euclidian geometry.png|thumb|upright=1.3|Three examples of different geometries:  [[Euclidean geometry|Euclidean]], [[elliptical geometry|elliptical]], and [[hyperbolic geometry|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 dimension|1-dimensional]] [[line (mathematics)|line]] <math>(l)</math> may be studied in isolation —in which case the ambient space of <math>l</math> is <math>l</math>, or it may be studied as an object embedded in [[Plane (mathematics)|2-dimensional]] [[Euclidean space]] <math>(\mathbb{R}^2)</math>—in which case the ambient space of <math>l</math> is <math>\mathbb{R}^2</math>, or as an object embedded in 2-dimensional [[hyperbolic space]] <math>(\mathbb{H}^2)</math>—in which case the ambient space of <math>l</math> is <math>\mathbb{H}^2</math>.  To see why this makes a difference, consider the statement "[[parallel (geometry)|Parallel]] lines never intersect."  This is true if the ambient space is <math>\mathbb{R}^2</math>, but false if the ambient space is <math>\mathbb{H}^2</math>, because the [[Geometry|geometric properties]] of <math>\mathbb{R}^2</math> are different from the geometric properties of <math>\mathbb{H}^2</math>.  All spaces are [[subset]]s of their ambient space.

==See also==
* [[Configuration space (mathematics)|Configuration space]]
* [[Geometric space]]
* [[Manifold]] and [[ambient manifold]]
* [[Submanifold]]s and [[Hypersurface]]s
* [[Riemannian manifolds]]
* [[Ricci curvature]]
* [[Differential form]]

==Further reading==
*{{cite book |first=W. H. A. |last=Schilders |first2=E. J. W. |last2=ter Maten |first3=Philippe G. |last3=Ciarlet |title=Numerical Methods in Electromagnetics |volume=Special Volume |publisher=[[Elsevier]] |year=2005 |isbn=0-444-51375-2 |pages=120ff }}
*{{cite book |author-link=Stephen Wiggins |first=Stephen |last=Wiggins |title=Chaotic Transport in Dynamical Systems |url=https://archive.org/details/chaotictransport0002wigg |url-access=registration |location=Berlin |publisher=Springer |year=1992 |isbn=3-540-97522-5 |pages=209ff }}

[[Category:Geometry]]
[[Category:Topology]]


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