Lecture 9: The Whitney Embedding Theorem (USTC Differentiable Manifolds)
by Zuoqin Wang · University of Science and Technology of China
Self-contained lecture note proving that any smooth m-manifold embeds in Euclidean space of dimension 2m+1. Traces the extrinsic-versus-intrinsic definition question from Riemann and Weyl, states the strong 1944 theorem and the Whitney trick, and surveys sharper dimension results.
More resources on Embedding Theorems
Wolfram MathWorld
MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.
nLab: Whitney Embedding Theorem
nLab reference entry on the Whitney embedding theorem, stating that every smooth n-manifold embeds in Euclidean space of dimension 2n. Gives the weak and strong versions, related immersion results, and pointers to original papers and textbook proofs for further study.
Lectures on Differential Topology
The best free full-course substitute for Guillemin and Pollack: complete proofs, lecture-by-lecture structure, and the same transversality-first organization, so a learner without book access loses nothing on the core material. Free 300-page lecture notes from Harvard Math 132 and Toronto MAT1300. Treats smooth maps and derivatives, immersions, submersions, embeddings, the Whitney embedding theorem, transversality and the preimage theorem, then intersection theory and Morse theory.
Embeddings and Immersions
The only book-length treatment found whose entire subject is this topic rather than a chapter within general differential topology; it carries a learner past the Whitney bound into the Haefliger classification and Gromov's h-principle machinery. Monograph devoted entirely to smooth embeddings and immersions, opening with the results of Whitney and Haefliger, then the Smale-Hirsch theorem and Gromov's convex integration theory. Two of its chapters treat embeddings of smooth manifolds directly; 183 pages, translated from Japanese.
nLab
Collaborative wiki for mathematics, physics and philosophy written from a category-theoretic viewpoint, with cross-linked entries on categories, functors, adjunctions, topos theory, higher category theory and homotopy theory. Readers can look up precise definitions, examples and references for advanced structural mathematics.