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manifolds and cobordisms
cobordism theory, Introduction
Definitions
Genera and invariants
Classification
Theorems
synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
Models
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
The (strong) Whitney embedding theorem states that every smooth manifold (Hausdorff and sigma-compact) of dimension has an embedding of smooth manifolds in the Euclidean space of dimension .
Notice that it is easy to see that every smooth manifold embeds into the Euclidean space of some dimension (this prop.). The force of Whitney’s strong embedding theorem is to find the lowest dimension that still works in general.
Analogous statements:
Riemannian manifolds and isometric embeddings: Nash embedding theorem
Named after Hassler Whitney.
Hassler Whitney, Differentiable manifolds, Ann. Math. 37, 645–680 (1936).
Lev Pontrjagin, Section 2.2 of: Smooth manifolds and their applications in Homotopy theory, Trudy Mat. Inst. im Steklov, No 45, Izdat. Akad. Nauk. USSR, Moscow, 1955 (AMS Translation Series 2, Vol. 11, 1959) (doi:10.1142/9789812772107_0001)
See also
Wikipedia, Whitney embedding theorem
Paul Rapoport, Introduction to Immersion, Embeddingand the Whitney Embedding Theorem, 2015 (pdf)
Last revised on December 26, 2024 at 22:37:15. See the history of this page for a list of all contributions to it.