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 ) {\displaystyle (l)} may be studied in isolation —in which case the ambient space of l {\displaystyle l} is the real line, or it may be studied as an object embedded in 2-dimensional Euclidean space ( R 2 ) {\displaystyle (\mathbb {R} ^{2})}—in which case the ambient space of l {\displaystyle l} is R 2 {\displaystyle \mathbb {R} ^{2}}, or as an object embedded in 2-dimensional hyperbolic space ( H 2 ) {\displaystyle (\mathbb {H} ^{2})}—in which case the ambient space of l {\displaystyle l} is H 2 {\displaystyle \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 R 2 {\displaystyle \mathbb {R} ^{2}}, but false if the ambient space is H 2 {\displaystyle \mathbb {H} ^{2}}, because the geometric properties (in particular, curvature) of R 2 {\displaystyle \mathbb {R} ^{2}} are different from the geometric properties of H 2 {\displaystyle \mathbb {H} ^{2}}. All spaces are subsets of their ambient space.

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Further reading