Geometry of Vector Sheaves: An Axiomatic Approach to Differential Geometry Volume II: Geometry. Examples and Applications (Mathematics and Its Applications (439))

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. This Two-volume Monograph Obtains Fundamental Notions And Results Of The Standard Differential Geometry Of Smooth (cinfinity) 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. By Anastasios Mallios. 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. Front Matter....Pages i-xxi Front Matter....Pages xxiii-xxiii Geometry of Vector Sheaves. A -connections....Pages 1-96 A -connections. Local Theory....Pages 97-183 Curvature....Pages 185-241 Characteristic Classes....Pages 243-275 Front Matter....Pages N1-N1 Classical Theory....Pages 277-298 Sheaves and presheaves with topological algebraic structures....Pages 299-386 Back Matter....Pages 387-438