Geometry of Vector Sheaves: An Axiomatic Approach to Differential Geometry Volume II: Geometry. Examples and Applications

ISBN-13: 9789401061025 / Angielski / Miękka / 2012 / 438 str.

Anastasios Mallios

Opis

This two-volume monograph obtains fundamental notions and results of the standard differential geometry of smooth (C^INFINITY) manifolds, without using differential calculus. Here, the sheaf-theoretic character is emphasised. This has theoretical advantages such as greater perspective, clarity and unification, but also practical benefits ranging from elementary particle physics, via gauge theories and theoretical cosmology (differential spaces'), to non-linear PDEs (generalised functions). Thus, more general applications, which are no longer smooth' in the classical sense, can be coped with. The treatise might also be construed as a new systematic endeavour to confront the ever-increasing notion that the world around us is far from being smooth enough'.
Audience: This work is intended for postgraduate students and researchers whose work involves differential geometry, global analysis, analysis on manifolds, algebraic topology, sheaf theory, cohomology, functional analysis or abstract harmonic analysis.

Spis treści

Part One: Vector Sheaves. General Theory. I. Sheaf Theory. II. II. Sheaves and Presheaves with Algebraic Structure. III. Sheaf Cohomology. IV. Linear and Multilinear Algebra of Vector Sheaves. V. Vector Sheaves and Cohomology. Appendix: Category Jargon. Bibliography. Notational Index. Subject Index. Part Two: Geometry. VI. Geometry of Vector Sheaves. A-Connections. VII. A-Connections. Local Theory. VIII. Curvature. IX. Characteristic Classes. Part Three: Examples and Applications. X. Classical Theory. XI. Sheaves and Presheaves with Topological Algebraic Structures. Bibliography. Notational Index. Subject Index.