Ambient space (mathematics)

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 the real line, 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 (in particular, curvature) 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
• Abstract variety and Projective embedding

Further reading

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