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.

Contents: Elements of Differential Calculus over Commutative Algebras:; Algebraic Tools; Smooth Manifolds; Vector Bundles; Vector Fields; Differential Forms; Lie Derivative; Basic Differential Calculus on Fat Manifolds:; Basic Definitions; The Lie Algebra of Der-operators; Fat Vector Fields; Fat Fields and Vector Fields on the Total Space; Induced Der-operators; Fat Trajectories; Inner Structures; Linear Connections:; Basic Definitions and Examples; Parallel Translation; Curvature; Operations with Linear Connections; Linear Connections and Inner Structures; Covariant Differential:; Fat de Rham Complexes; Covariant Differential; Compatible Linear Connections; Linear Connections Along Fat Maps; Covariant Lie Derivative; Gauge/Fat Structures and Linear Connections; Cohomological Aspects of Linear Connections:; An Introductory Example; Cohomology of Flat Linear Connections; Maxwell's Equations; Homotopy Formula for Linear Connections; Characteristic Classes.