Whitney embedding theorem

The theorem that every smooth manifold of dimension n admits a smooth embedding into Euclidean space of dimension 2n. Its weak form guarantees an embedding into dimension 2n+1.

Smooth manifold: A space locally modeled on Euclidean space whose coordinate changes are smooth. The theorem applies to smooth manifolds and measures its bound using their dimension.
Hassler Whitney: An American mathematician who made foundational contributions to topology, differential geometry, and singularity theory. Whitney proved the embedding theorem and developed the surrounding theory of manifolds.
Weak Whitney embedding theorem: The theorem that every smooth n-manifold embeds in Euclidean space of dimension 2n+1. This version gives a simple universal bound one dimension higher than Whitney's strong theorem.
Submanifold: A subset of a manifold that is itself a manifold with compatible smooth structure. An embedded manifold is realized as a submanifold of Euclidean space.
Smooth embedding: A smooth map that is an immersion and a homeomorphism onto its image. The theorem guarantees this stronger, non-self-intersecting placement in Euclidean space.
Differential topology: The study of smooth manifolds and smooth maps between them. The theorem became a foundational result in this field.
Nash embedding theorem: The theorem that every Riemannian manifold admits an isometric embedding into some Euclidean space. Nash requires preservation of the metric, unlike Whitney's theorem, and has different dimension bounds.
Tubular neighborhood theorem: The theorem that a neighborhood of a smooth submanifold is modeled by a neighborhood of its normal bundle. Once embedded, a manifold can be studied through its surrounding normal directions.
General position: A condition in which geometric objects intersect transversely, with dimensions controlling whether intersections occur. General-position arguments explain why a sufficiently high-dimensional target permits embeddings.
Whitney immersion theorem: The theorem that every smooth n-manifold immerses in Euclidean space of dimension 2n−1. Whitney's immersion result complements the embedding theorem with a sharper bound allowing self-intersections.

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Johnson–Lindenstrauss lemmaCompared with: Whitney’s theorem embeds entire manifolds; this lemma compresses a finite set of points.