The convenient setting of global analysis. (English) Zbl 0889.58001

Kriegl, Andreas; Michor, Peter W.

Mathematical Surveys and Monographs. 53. Providence, RI: American Mathematical Society (AMS). x, 618 p. (1997).

The book is devoted to the study of manifolds modeled on locally convex spaces which are smooth in an appropriate sense. To this aim a generalized differential calculus is developed in the setting of locally convex spaces. The authors’ approach relies on the following idea: a mapping is called smooth if it maps smooth curves to smooth curves. The rules for such a differential calculus are established and carefully compared with the classical results. On this basis the needed tools are constructed: smooth partitions of unity, the foundations of manifold theory, the relation between tangent vectors and derivations, vector fields, differential forms, cohomology, Lie groups, bundles. The main motivation and applications of these extensions concern the infinitedimensional differential geometry and global analysis. Specifically, the authors discuss manifolds of mappings, groups of diffeomorphisms, Riemannian metrics, representations of Lie groups, perturbation theory of operators. The book includes the authors’ original contributions in the field.

Reviewer: D.Motreanu (Iaşi)

MSC:

58-02	Research exposition (monographs, survey articles) pertaining to global analysis
58C25	Differentiable maps on manifolds
46A03	General theory of locally convex spaces
58B10	Differentiability questions for infinite-dimensional manifolds
58D15	Manifolds of mappings

Keywords:

manifold; differentiability; Riemannian metric; bundle; smooth mapping; locally convex spaces; differential calculus; Lie groups