Fat manifolds and linear connections

The theory of connections is central not only in pure mathematics (differential and algebraic geometry), but also in mathematical and theoretical physics (general relativity, gauge fields, mechanics of continuum media).

The now-standard approach to this subject was proposed by Ch.

Ehresmann 60 years ago, attracting first mathematicians and later physicists by its transparent geometrical simplicity.

Unfortunately, it does not extend well to a number of recently emerged situations of significant importance (singularities, supermanifolds, infinite jets and secondary calculus, etc.).

Moreover, it does not help in understanding the structure of calculus naturally related with a connection.In this unique book, written in a reasonably self-contained manner, the theory of linear connections is systematically presented as a natural part of differential calculus over commutative algebras.

This not only makes easy and natural numerous generalizations of the classical theory and reveals various new aspects of it, but also shows in a clear and transparent manner the intrinsic structure of the associated differential calculus.

The notion of a “fat manifold” introduced here then allows the reader to build a well-working analogy of this “connection calculus” with the usual one.Sample Chapter(s)Foreword (75 KB)Chapter 0: Elements of Di erential Calculus over Commutative Algebras (636 KB)Contents: Elements of Differential Calculus over Commutative Algebras:Algebraic ToolsSmooth ManifoldsVector BundlesVector FieldsDifferential FormsLie DerivativeBasic Differential Calculus on Fat Manifolds:Basic DefinitionsThe Lie Algebra of Der-operatorsFat Vector FieldsFat Fields and Vector Fields on the Total SpaceInduced Der-operatorsFat TrajectoriesInner StructuresLinear Connections:Basic Definitions and ExamplesParallel TranslationCurvatureOperations with Linear ConnectionsLinear Connections and Inner StructuresCovariant Differential:Fat de Rham ComplexesCovariant DifferentialCompatible Linear ConnectionsLinear Connections Along Fat MapsCovariant Lie DerivativeGauge/Fat Structures and Linear ConnectionsCohomological Aspects of Linear Connections:An Introductory ExampleCohomology of Flat Linear ConnectionsMaxwell's EquationsHomotopy Formula for Linear ConnectionsCharacteristic ClassesReadership: Advanced undergraduate and graduate students and researchers in mathematics, mathematical and theoretical physics.