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Embeddings and immersions. Translated from the Japanese by Kiki Hudson. (English) Zbl 0810.57001
Translations of Mathematical Monographs. 124. Providence, RI: American Mathematical Society (AMS). x, 183 p. (1993).
[The Japanese original was published by Iwanami Shoten, Tokyo (1984).]
The object of this monograph is to introduce the reader with a basic knowledge of topology and analysis to the theory of smooth immersions and imbeddings. By way of motivation the author starts with Whitney’s study of regular closed curves in the plane, and then moves on to a fairly detailed presentation of Whitney’s main classical results on smooth immersions and imbeddings of smooth manifolds into \(R^{2n}\) and \(R^{2n + 1}\). The presentation follows the original treatment closely. The main body of the monograph is devoted to the Smale-Hirsch-Phillips- Gromov work on immersions. He proves Gromov’s theorem first and then deduces the other results as corollaries. Then he takes up Gromov’s alternative treatment based on the theory of convex integration of differential relations. In the final chapters, as applications of the previous theory, he gives Haefliger’s classification of foliations on open manifolds and a study of complex structures, also on open manifolds.

57-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to manifolds and cell complexes
57R40 Embeddings in differential topology
57R42 Immersions in differential topology
57R30 Foliations in differential topology; geometric theory
57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
58D10 Spaces of embeddings and immersions