Ebook: Handbook of Normal Frames and Coordinates

This book provides the first comprehensive and complete overview on results and methods concerning normal frames and coordinates in differential geometry, with emphasis on vector and differentiable bundles.

The book can be used as a reference manual, for reviewing the existing results and as an introduction to some new ideas and developments. Virtually all essential results and methods concerning normal frames and coordinates are presented, most of them with full proofs, in some cases using new approaches.

All classical results are expanded and generalized in various directions. For example, normal frames and coordinates are defined and investigated for different kinds of derivations, in particular for (possibly linear) connections on manifolds, with or without torsion, in vector bundles and on differentiable bundles; they are explored also for (possibly parallel) transports along paths in vector bundles. Theorems of existence, uniqueness and, possibly, holonomicity of normal frames and coordinates are proved; mostly, the proofs are constructive and some of their parts can be used independently for other tasks.

Numerous examples and exercises illustrate the material. Graduate students and researchers alike working in differential geometry or mathematical physics will benefit from this resource of ideas and results which are of particular interest for applications in the theory of gravitation, gauge theory, fibre bundle versions of quantum mechanics, and (Lagrangian) classical and quantum field theories.

The main subject of the book is an up-to-date and in-depth survey of the theory of normal frames and coordinates in differential geometry.

The book can be used as a reference manual, a review of the existing results and an introduction to some new ideas and developments.

Practically all existing essential results and methods concerning normal frames and coordinates can be found in the book. Most of the results are represented in detail with full, in some cases new, proofs. All classical results are expanded and generalized in various directions. The normal frames and coordinates, for example, are defined and investigted for different kinds of derivations, in particular for (possibly linear) connections (with or without torsion) on manifolds, in vector bundes and on differentiable bundles; they are explored also for (possibly parallel) transports along paths in vector bundles. Theorems of existence, uniqueness and, possibly, holonomicity of the normal frames and coordinates are proved; mostly, the proofs are constructive and some of their parts can be used independently for other tasks.

Besides published results, their extensions and generalizations, the book contains completely new results which appear for the first time, such as for instance some links between (existence of) normal frames/coordinates in vector bundles and curvature/torsion.

As secondary items, elements of the theory of (possibly linear) connections on manifolds, in vector bundles and on differentiable bundles and of (possibly parallel or linear) transports along paths in vector and on differentiable bundles are presented.

The theory of the monograph is illustrated with a number of examples and exercices.

The contents of the book can be used for applications in differential geometry, e.g. in the theories of (linear) connections and (linear or parallel) transports along paths, and in the theoretical/mathematical physics, e.g. in the theories of gravitation, gauge theories and fibre bundle versions of quantum mechanics and (Lagrangian) classical and quantum field theories.

The potential audience ranges from graduate and postgraduate students to research scientists working in the fields of differential geometry and theoretical/mathematical physics.