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. Practically all existing essential results and methods concerning normal frames and coordinates can be found in the book. Most of the results are presented in detail with full, and in some cases new, proofs. A large number of examples and exercises illustrate the material. 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. 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.

Inhaltsverzeichnis

Manifolds, Normal Frames and Riemannian Coordinates.- Existence, Uniqueness and Construction of Normal Frames and Coordinates for Linear Connections.- Normal Frames and Coordinates for Derivations on Differentiable Manifolds.- Normal Frames in Vector Bundles.- Normal Frames for Connections on Differentiable Fibre Bundles.

Zusammenfassung

The main subject of this book is an up-to-date and in-depth survey of the theory of normal frames and coordinates in di?erential geometry. The existing results, as well as new ones obtained lately by the author, on the theme are presented. The text is so organized that it can serve equally well as a reference manual, introduction to and review of the current research on the topic. Correspondingly, the possible audience ranges from graduate and post-graduate students to sci- tists working in di?erential geometry and theoretical/mathematical physics. This is re?ected in the bibliography which consists mainly of standard (text)books and journal articles. The present monograph is the ?rst attempt for collecting the known facts concerting normal frames and coordinates into a single publication. For that r- son, the considerations and most of the proofs are given in details. Conventionally local coordinates or frames, which can be holonomic or not, are called normal if in them the coe?cients of a linear connection vanish on some subset, usually a submanifold, of a di?erentiable manifold. Until recently the ex- tence of normal frames was known (proved) only for symmetric linear connections on submanifolds of a manifold. Now the problems concerning normal frames for derivationsof thetensor algebraovera di?erentiablemanifoldarewellinvestigate; in particular they completely cover the exploration of normal frames for arbitrary linear connections on a manifold. These rigorous results are important in conn- tion with some physical applications.

Zusatztext

From the reviews:
“This is a careful and rather comprehensive presentation of theory and methods related to normal frames in differential geometry. … The book is carefully written, provides a good overview, and contains many interesting aspects not available otherwise. It is a valuable addition to the literature on connections in differential geometry.” (M. Kunzinger, Monatshefte für Mathematik, Vol. 154 (1), May, 2008)

Bericht

From the reviews:
"This is a careful and rather comprehensive presentation of theory and methods related to normal frames in differential geometry. ... The book is carefully written, provides a good overview, and contains many interesting aspects not available otherwise. It is a valuable addition to the literature on connections in differential geometry." (M. Kunzinger, Monatshefte für Mathematik, Vol. 154 (1), May, 2008)

Produktdetails

• Autoren: Bozhidar Z Iliev, Bozhidar Z. Iliev
• Verlag: Springer, Basel
• Sprache: Englisch
• Produktform: Fester Einband
• Erschienen: 20.09.2006
• EAN: 9783764376185
• ISBN: 978-3-7643-7618-5
• Seiten: 441
• Gewicht: 912 g
• Illustration: XV, 441 p.
• Serien: Progress in Mathematical Physics; Progress in Mathematical Physics
• Themen: Naturwissenschaften, Medizin, Informatik, Technik > Mathematik > Geometrie; Sachbuch > Natur, Technik > Astronomie: Allgemeines, Nachschlagewerke; B, Mathematics and Statistics, Manifold, Differential Geometry, vector bundle, normal frame