Ambient Space (mathematics)

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

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