Misplaced Pages

This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. Please help improve this article by introducing more precise citations. (May 2024) (Learn how and when to remove this message)

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 ${\displaystyle (l)}$ may be studied in isolation —in which case the ambient space of ${\displaystyle l}$ is ${\displaystyle l}$, or it may be studied as an object embedded in 2-dimensional Euclidean space ${\displaystyle (\mathbb {R} ^{2})}$—in which case the ambient space of ${\displaystyle l}$ is ${\displaystyle \mathbb {R} ^{2}}$, or as an object embedded in 2-dimensional hyperbolic space ${\displaystyle (\mathbb {H} ^{2})}$—in which case the ambient space of ${\displaystyle l}$ is ${\displaystyle \mathbb {H} ^{2}}$. To see why this makes a difference, consider the statement "Parallel lines never intersect." This is true if the ambient space is ${\displaystyle \mathbb {R} ^{2}}$, but false if the ambient space is ${\displaystyle \mathbb {H} ^{2}}$, because the geometric properties of ${\displaystyle \mathbb {R} ^{2}}$ are different from the geometric properties of ${\displaystyle \mathbb {H} ^{2}}$. 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

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

This geometry-related article is a stub. You can help Misplaced Pages by expanding it.

• v
• t
• e

• Geometry
• Topology
• Geometry stubs