de Rham, Georges [Chern, S. S.] 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. Formes, courants, formes harmoniques. Paris: Hermann (1955; Zbl 0065.32401)]. 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 are 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: Wolfgang Kühnel (Stuttgart) Cited in 2 ReviewsCited in 153 Documents MathOverflow Questions: A text about Schwartz distributions in vector bundles MSC: 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 Keywords:Stokes formula; double forms; double currents; regularizaton; Kronecker index; polyhedral subdivision; covariant derivative; geodesic distance; parametrix; regularity of harmonic currents; square summable harmonic forms Citations:Zbl 0065.32401 × Cite Format Result Cite Review PDF