Balan V., Rahula M., Voicu N. Tangent Structures in Geometry and Their Applications

Contents

Abstract

This is a scientific monograph on modern differential geometry with applications to the mechanics of continuous media. It is recommended to researchers, but it can also serve as a manual for undergraduate, graduate or PhD students. In the first part of the book (Chapters 1-6), the main geometric structures: tangent functor and its iterations (multiple sector bundles, levels), Lie-Cartan calculus in non-holonomic basis, connections in bundles, exponential law in infinite jet space, higher order movements, are presented in an original manner. In the second part (Chapters 7-13), some remarkable topics on the tangent bundle are elaborated: theory of geodesics in the second order geometry, generalized gauge theory, Einstein-Yang-Mills equations, together with Jacobi stability and KCC-theory.

Introduction

This book is divided into 13 Chapters dedicated to different topics of the actual Global Analysis and Differential Geometry, like: differential prolongations, Lie-Cartan calculus, connections in bundles, the geometry of differential equations and applications to Theoretical Physics. Consequently each Chapter can be read independently and, thanks to this, the reader will conveniently focus on the topic, and subsequently better assimilate the exposed subject. The main leitmotif of the book are movements, transformations of movements and movements of movements – i.e., higher-order movements. By movement, we generally understand an arbitrary process, which continuously varies in time. The first linear approximation of the process provides the differential of mapping. There holds a linearization: the process stops at a given moment, displacements of points are replaced by infinitesimal translations and the analysis of the process reduces to algebraic actions, in particular, to operations with vector fields and Lie differentiations. In the framework of a united system, we propose a multitude of original ideas related, for instance, to the theory of geodesics, field theory in pseudo-Finsler spaces, gauge theory, Einstein–Yang-Mills equations, Jacobi stability or KCC,theory, and numerous illustrative complementary exercises.

We further include a brief outlook on the main topics:

Chapter 1. Linearization. Tangent functor. Higher order tangent bundles are iteratively built by means of the tangent functor $T$. As a generalization of classical differential forms, White's theory of sector-forms is developed.

Chapter 2. Tangent groups. The functor $T$ allows one to discuss tangent groups of Lie groups and to build a group representation theory. It is proposed a new approach to applicative issues – in particular, to gauge theory.

Chapter 3. Lie-Cartan calculus in nonholonomic basis. Lie derivatives and Cartan forms are dual concepts in a united calculus approach. The development relies on a nonholonomic basis, which allows us to simultaneously review contemporary global analysis and classical theory.

Chapter 4. Connections in bundles. The notion of connection plays a central role not only in tensor analysis, but other fields as well. The topical issue of higher-order connections requires a special attention.

Chapter 5. Jets and exponential law. A nonholonomic jet (by Ehresmann) is, in its essence, the set of coefficients of some sector-form. The theory of algebraic and differential invariants entirely emerges in the framework of exponential law.

Chapter 6. Analytical mechanics. Hamilton formalism. In the theory of higher-order motions, a determinant role is played by tangent and osculating bundles. Any Hamiltonian system as a section of the tangent bundle $T^2M$ reduces to a Lagrangian system on the osculating bundle Osc$M$.

Chapter 7. Covariant differentiation. Osculating bundle. On these fibered bundles, Ehresmann connections and associated d-connections related to some classical variational problems are determined.

Chapter 8. Geodesic deviations. A special attention is paid to geodesic deviation and its description in terms of second order tangent bundle geometry.

Chapter 9. Finsler-type structures. Finsler Geometry is an essential ingredient for modeling anisotropic phenomena from Mechanics and Relativity. The main specific geometric tools are described and their proper-Finslerian features are presented.

Chapter 10. Structural stability. Structural stability (KCC), specific to second-order DE written in canonical form, has recently found numerous applications in various fields – ecology, biology, seismology, etc. The features of this theory, its relation to the Finslerian framework and its major applications are described.

Chapter 11. Electromagnetic field theory in Finsler spaces. The classical theory of electromagnetic field in curved spaces is extended to the case when the metric is direction-dependent (in particular, of Finsler type).

Chapter 12. Generalized gauge framework. In the attempt of unifying gravity and electromagnetism, numerous alternative models consider as main ingredient metric structures on Osc$^kM$. The specific geometric objects of the generalized gauge Finslerian theory are presented, and further, using the Hilbert-Palatini variational principle, the generalized Einstein-Yang-Mills equations are derived

Appendix A. Decomposition of a map and the rank problem.

Appendix B. Lie derivatives in a nonholonomic basis.

Appendix C. Brief overview on $m$-th root structures.

Bibliography

Index

Description in brief:

An invariant (coordinate-free) apparatus based on Lie derivatives and Cartan forms in a nonholonomic basis is elaborated. The theory of connections on general bundles and, in particular, linear connections on vector and tangent bundles is developed. The tangent and osculating structures are considered in detail. Finsler Geometry is an essential ingredient for modeling anisotropic phenomena from Mechanics and Relativity. We describe the main specific geometric tools and present their proper-Finslerian features. In the attempt of unifying gravity and electromagnetism, we consider numerous alternative models, having as main ingredient metric structures on the tangent bundle. The geometric objects of the generalized gauge Finslerian theory are studied.

The book is of interest for:

1) researchers wanting to be acquainted with coordinate-free analysis (derivatives, integrals, differential equations), i.e., to Lie-Cartan calculus;

2) applied scientists (physicists, specialists in mechanics) dealing with theoretical problems of mechanics of continuous media, quantum mechanics etc;

3) authors of monographs and textbooks aspiring to provide an elegant presentation of similar subjects;

4) readers interested in philosophical questions such as higher order motions, laws of motion, their relativity and stability.

The following themes can be taken separately and used as didactic material:

– invariant Lie-Cartan calculus;

– theory of algebraic and differential invariants, singularities of mappings

(catastrophes);

– geometric theory of differential equations;

– Lagrangian Mechanics and Finsler Geometry.

The reader is advised:

– to have knowledge of general Mathematics university courses,

– to be skilled in handling formulas (writing formulas in matrix form, indexing, derivation relative to vector fields, calculation of flows),

– to independently solve the suggested exercises and to properly integrate them in the general framework,

– to efficiently make use of the reference list.

Citations to titles in references and accompanying remarks are provided within text, while the presentation is running.

The text contains numerous exercises, which are frequently referred. As well, we shall denote, e.g., by 3.4.1: Section (Chapter) 3, Subsection 4, Subsubsection 1.

The authors are grateful for the financial help offered by the Mathematics Institute of the University of Tartu, by the Mathematical Department of University Politehnica of Bucharest and by University "Transilvania" of Brasov. The present work was developed under the auspices of the Romanian Academy International Cooperation Grant GAR 6/2010-2011, was supported by the Sectorial Operational Program Human Resources Development (SOP HRD), financed from the European Social Fund and by Romanian Government under the Project number POSDRU/89/1.5/S/59323, and from the Estonian Targeted Financing Project SF0180039s08.

We address our thanks to the Editors Armand Colin - Paris (Mr. Antoine Bonfait), for giving us the permission to use, as illustrations of our monograph, the vignettes from the book "L'id\'ee fixe du Savant Cosinus", authored by Cristophe.

We conclude by announcing that recently has appeared the new edition of the monograph Lectures on Differential Geometry by S.S. Chern, W.H. Chen and K.S. Lam, World Scientific - 2011, which represents an exquisite complementary text to our volume.

About the authors