Differentiable manifolds. Forms, currents, harmonic forms. Transl. from the French by F. R. Smith. Introduction to the English ed. by S. S. Chern. (English) Zbl 0534.58003

Grundlehren der Mathematischen Wissenschaften, 266. Berlin etc.: Springer-Verlag. X, 167 p. DM 78.00; $ 28.40 (1984).
This is a very welcome translation of the french original [Variétés Différentiables (1955; Zbl 0065.324)]. The introduction of the original edition has been slightly modified. The new version contains the description of the content whereas the history of the problem and of the book itself is explained in a ”Préface á l’édition anglaise” by the author and in an ”Introduction to the English Edition” by S. S. Chern. In particular the author’s preface contains some autobiographical notes, and Chern’s introduction explains the link of the present book with work of W. V. D. Hodge, H. Weyl, H. Cartan, J.-P. Serre and others. The text of the book itself has not been changed except that the open question at the end of § 34 concerning harmonic forms on smooth but non-analytic manifolds has been answered and that there has been added a very last section § 35 on square summable harmonic forms.
Reviewer: W.Kühnel


58Axx General theory of differentiable manifolds
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
55Nxx Homology and cohomology theories in algebraic topology
58A05 Differentiable manifolds, foundations
58A10 Differential forms in global analysis
58A12 de Rham theory in global analysis
58A14 Hodge theory in global analysis
58A25 Currents in global analysis


Zbl 0065.324