Ebook: Geometry of Principal Sheaves
- Tags: Differential Geometry
- Series: Mathematics and Its Applications 578
- Year: 2005
- Publisher: Springer Netherlands
- Edition: 1
- Language: English
- pdf
From the reviews of the first edition:
"The book provides an accessible, well-written monograph devoted to the theory of principal sheaves and their connections in the setting initiated by, Geometry of vector sheaves. … It is designed also as a reference book with detailed expositions and complete and self-contained proofs." (Witold Mozgawa, Zentralblatt MATH, Vol. 1082, 2006)
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.
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.
Content:
Front Matter....Pages i-xvi
Sheaves and all that....Pages 1-52
The category of differential triads....Pages 53-87
Lie sheaves of groups....Pages 89-129
Principal sheaves....Pages 131-161
Vector and associated sheaves....Pages 163-210
Connections on principal sheaves....Pages 211-254
Connections on vector and associated sheaves....Pages 255-277
Curvature....Pages 279-340
Chern-Weil theory....Pages 341-368
Applications and further examples....Pages 369-407
Back Matter....Pages 409-444
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.
Content:
Front Matter....Pages i-xvi
Sheaves and all that....Pages 1-52
The category of differential triads....Pages 53-87
Lie sheaves of groups....Pages 89-129
Principal sheaves....Pages 131-161
Vector and associated sheaves....Pages 163-210
Connections on principal sheaves....Pages 211-254
Connections on vector and associated sheaves....Pages 255-277
Curvature....Pages 279-340
Chern-Weil theory....Pages 341-368
Applications and further examples....Pages 369-407
Back Matter....Pages 409-444
....