Introduction to my book “Advanced Differential Geometry for Theoreticians”

G.Sardanashvily, Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory (Lambert Academic Publishing, Saarbrucken, 2013)  #

In contrast with quantum field theory, classical field theory can be formulated in a strict mathematical way by treating classical fields as sections of smooth fibre bundles. This also is the case of time-dependent non-relativistic mechanics on fibre bundles over R. This book aim to compile the relevant material on fibre bundles, jet manifolds, connections, graded manifolds and Lagrangian theory. The book is based on the graduate and post graduate courses of lectures given at the Department of Theoretical Physics of Moscow State University (Russia). It addresses to a wide audience of mathematicians, mathematical physicists and theoreticians. It is tacitly assumed that the reader has some familiarity with the basics of differential geometry.

Contents

1 Geometry of fibre bundles: 1.1 Fibre bundles, 1.2 Vector and affine bundles, 1.3 Vector fields, 1.4 Exterior and tangent-valued forms.

2 Jet manifolds: 2.1 First order jet manifolds, 2.2 Higher order jet manifolds, 2.3 Differential operators and equations, 2.4 Infinite order jet formalism.

3 Connections on fibre bundles: 3.1 Connections as tangent-valued forms, 3.2 Connections as jet bundle sections, 3.3 Curvature and torsion, 3.4 Linear and affine connections, 3.5 Flat connections, 3.6 Connections on composite bundles.

4 Geometry of principal bundles: 4.1 Geometry of Lie groups, 4.2 Bundles with structure groups, 4.3 Principal bundles, 4.4 Principal connections, 4.5 Canonical principal connection, 4.6 Gauge transformations, 4.7 Geometry of associated bundles, 4.8 Reduced structure.

5 Geometry of natural bundles: 5.1 Natural bundles, 5.2 Linear world connections, 5.3 Affine world connections.

6 Geometry of graded manifolds: 6.1 Grassmann-graded algebraic calculus, 6.2 Grassmann-graded differential calculus, 6.3 Graded manifolds, 6.4 Graded differential forms.

7 Lagrangian theory: 7.1 Variational bicomplex, 7.2 Lagrangian theory on fibre bundles, 7.3 Grassmann-graded Lagrangian theory, 7.4 Noether identities, 7.5 Gauge symmetries.

8 Topics on commutative geometry: 8.1 Commutative algebra, 8.2 Differential operators on modules, 8.3 Homology and cohomology of complexes, 8.4 Differential calculus over a commutative ring, 8.5 Sheaf cohomology, 8.6 Local-ringed spaces.