Ambient space (mathematics)

File:Euclidian and non euclidian geometry.png

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 $l$, 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 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
• Riemannian manifolds
• Ricci curvature
• Differential form

Further reading

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