Vassiliou, Efstathios. creator author. Hazewinkel, M. editor. SpringerLink (Online service) text ne 2005 monographic eng

electronic

electronic resource

access XVI, 444 p. online resource. The book provides a detailed introduction to the theory of connections on principal sheaves in the framework of Abstract Differential Geometry (ADG). This is a new approach to differential geometry based on sheaf theoretic methods, without use of ordinary calculus. This point of view complies with the demand of contemporary physics to cope with non-smooth models of physical phenomena and spaces with singularities. Starting with a brief survey of the required sheaf theory and cohomology, the exposition then moves on to differential triads (the abstraction of smooth manifolds) and Lie sheaves of groups (the abstraction of Lie groups). Having laid the groundwork, the main part of the book is devoted to the theory of connections on principal sheaves, incorporating connections on vector and associated sheaves. Topics such as the moduli sheaf of connections, classification of principal sheaves, curvature, flat connections and flat sheaves, Chern-Weil theory, are also treated. The study brings to light fundamental notions and tools of the standard differential geometry which are susceptible of the present abstraction, and whose role remains unexploited in the classical context, because of the abundance of means therein. However, most of the latter are nonsensical in ADG. Sheaves and all that -- The category of differential triads -- Lie sheaves of groups -- Principal sheaves -- Vector and associated sheaves -- Connections on principal sheaves -- Connections on vector and associated sheaves -- Curvature -- Chern-Weil theory -- Applications and further examples. by Efstathios Vassiliou ; edited by M. Hazewinkel. MATHEMATICS GLOBAL DIFFERENTIAL GEOMETRY MATHEMATICS DIFFERENTIAL GEOMETRY 516.36 9781402034169 99781402034169 http://dx.doi.org/10.1007/1-4020-3416-4 http://dx.doi.org/10.1007/1-4020-3416-4 100301 20260521092054.0 978-1-4020-3416-9