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Differentiable Manifolds - Forms, Currents, Harmonic Forms (Paperback, Softcover reprint of the original 1st ed. 1984)
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Differentiable Manifolds - Forms, Currents, Harmonic Forms (Paperback, Softcover reprint of the original 1st ed. 1984)
Series: Grundlehren der mathematischen Wissenschaften, 266
Expected to ship within 10 - 15 working days
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In this work, I have attempted to give a coherent exposition of the
theory of differential forms on a manifold and harmonic forms on a
Riemannian space. The concept of a current, a notion so general
that it includes as special cases both differential forms and
chains, is the key to understanding how the homology properties of
a manifold are immediately evident in the study of differential
forms and of chains. The notion of distribution, introduced by L.
Schwartz, motivated the precise definition adopted here. In our
terminology, distributions are currents of degree zero, and a
current can be considered as a differential form for which the
coefficients are distributions. The works of L. Schwartz, in
particular his beautiful book on the Theory of Distributions, have
been a very great asset in the elaboration of this work. The reader
however will not need to be familiar with these. Leaving aside the
applications of the theory, I have restricted myself to considering
theorems which to me seem essential and I have tried to present
simple and complete of these, accessible to each reader having a
minimum of mathematical proofs background. Outside of topics
contained in all degree programs, the knowledge of the most
elementary notions of general topology and tensor calculus and
also, for the final chapter, that of the Fredholm theorem, would in
principle be adequate.
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