Tangent Structures in Geometry and Their Applications

Description

Introduction
1 Linearization. Tangent functor
1.1 Fundamental categories
1.2 Representative functors
1.3 The tangent functor
1.4 Coordinates
1.5 Levels and sector-forms
1.6 Osculating bundles
2 Tangent groups
2.1 Leibniz rule
2.2 The tangent group
2.3 Elements of representation theory
2.4 Adjoint representation
2.5 Gauge groups
3 Lie-Cartan calculus in nonholonomic basis
3.1 Vector fields and flows
3.2 Lie derivatives
3.3 Nonholonomic basis
3.4 Entrainment of the basis
3.4.1 Invariant basis: the case C = 0
3.4.2 Linear flow: the case C’ = 0
3.4.3 Classification of linear flows
3.5 Symmetries of DE
3.5.1 Symmetry and integrating factor
3.5.2 Solutions and integrals of DE
3.6 Projecting vector field
3.7 Projectable vector field
3.7.1 Projection to a plane
3.7.2 Spray
3.8 Linear groups
3.8.1 Group GL(2,R)
3.8.2 The Lie algebra gl(2,R)
3.8.3 Generalization of rotation group
3.8.4 The group GL(3,R)
3.8.5 Theory of moments
4 Connections in bundles
4.1 Distribution on the manifold
4.2 Solutions and integrals
4.3 Symmetries of distribution
4.4 Infinitesimal symmetries
4.5 Integrating matrix
4.6 Bundles
4.6.1 General bundle
4.6.2 Vector bundle
4.6.3 Tangent bundle
4.7 The structure Deltah \oplus Deltav
4.7.1 Horizontal distribution Deltah
4.7.2 Specialized basis
4.7.3 Adapted basis
4.7.4 Symmetries of the distribution Deltah
4.7.5 Arbitrariness in the choice of the connection
4.7.6 Linear connection
4.7.7 Affine connection
4.7.8 Transformation of the adapted basis
4.7.9 Symmetry and integrating matrix
4.8 Morphism of bundles with connection
4.8.1 Invariant blocks of the Jacobian matrix
4.8.2 The object Falphaij
4.8.3 Review of connection theory
4.9 Connection in a double bundle
4.9.1 Specialized basis
4.9.2 Adapted basis
4.9.3 Linear connection
4.9.4 Orthogonal connection in a double bundle
4.10 Introduction to Miron-Atanasiu theory
4.11 Connection in a twofold bundle
5 Jets and exponential law
5.1 Exponential law
5.1.1 Implication I: invariants
5.1.2 Implication II: Cartan forms
5.1.3 Implication III: invariant basis
5.1.4 Implication IV: Lie fields
5.1.5 Implication V: the Lie form
5.2 The space Jn,m
5.2.1 Multi-indices
5.2.2 Multi-dimensional time
5.2.3 Embedding in Jn,m
5.3 Differentiation in the case of jet composition
5.3.1 Problem statement
5.3.2 Total differential group
5.4 Geometry of differential equations
5.4.1 Arbitrariness for DE solutions
5.4.2 Cartan distribution
5.4.3 DE applications
6 Analytical Mechanics
6.1 Lagrange, Legendre, Hamilton
6.1.1 Hamiltonian and Lagrangian equations
6.1.2 Hamiltonian formalism
6.1.3 Hamilton-Jacobi Theorem
6.1.4 Variational operators
6.2 Kepler and Newton
6.2.1 Center of gravity
6.2.2 The law of equal areas
6.2.3 Rotating pendulum
6.2.4 Three-body problem
7 Covariant differentiation. Osculating bundle
7.1 Ehresmann connections revisited
7.2 N-connections on the total space of a vector bundle
7.2.1 Definition and properties
7.2.2 Torsion and curvature
7.2.3 Exterior derivative
7.2.4 Deflection tensor. Berwald type N-connections
7.2.5 Metrical N-connections
7.3 N-connections on the tangent bundle
7.4 N-connections on double bundles
7.5 Osculating bundle
8 Geodesic deviations
8.1 Generalized Jacobi equation
8.1.1 Geometrical structures
8.1.2 Deduction of the equation
8.1.3 Examples
8.2 Geodesics in metric geometry
8.2.1 First variation of the energy. Geodesic equation
8.2.2 Second variation of the energy. Geodesic deviation
8.3 Geodesic deviation and connections on T2M
8.3.1 The second tangent bundle T2M
8.3.2 Lifts of curves and of families of curves to T2M
8.3.3 A special Ehresmann connection on T2M
9 Finsler-type structures
9.1 The jet context – basic structures
9.2 The Finsler structure – brief history and definitions
9.3 Notable examples of Finsler norms
9.4 The indicatrix of a Finsler space
9.4.1 Curvature of the Finsler indicatrix
9.4.2 The Berwald-Moór case
9.5 Finsler connections
9.6 The Finslerian model of gravitation
9.7 Finslerian m-th root structures
9.8 The Legendre transform of the Finsler space Hn
9.9 The Berwald-Moór dual structure
9.10 Hamilton vs. Lagrange equations
9.11 Geometric structures on T*Hn
10 Structural stability
10.1 Second order SODE. Structural stability
10.2 The Finslerian case
10.3 Overview and applications of the KCC theory
11 Electromagnetic field theory in Finsler spaces
11.1 A brief overview of the Riemannian case
11.1.1 Distances, volumes, divergence, codifferential
11.1.2 4-potential and electromagnetic tensor
11.1.3 Lagrangian, equations of motion and the second pair of Maxwell equations
11.1.4 Energy-momentum tensor
11.2 Some geometric structures in Finsler spaces
11.3 4-potential 1-form
11.4 Faraday 2-form and homogeneous Maxwell equations
11.5 Inhomogeneous Maxwell equations
11.6 Continuity equation and gauge invariance
11.7 Equations of motion
11.8 Stress-energy-momentum tensor
11.8.1 The case of flat pseudo-Finsler spaces
11.8.2 In general pseudo-Finsler spaces
11.9 Conclusion
12 Generalized gauge framework
12.1 Generalized gauge framework on M
12.1.1 Generalized gauge transformations and gauge covariant derivatives on M
12.1.2 Einstein – Yang-Mills equations on M
12.1.3 Quasi-metric gauge linear N-connections and Einstein – Yang-Mills equations on M
12.2 Generalized gauge transformations on OscM
12.2.1 Gauge transformations on OscM
12.2.2 Gauge covariant derivatives on OscM
12.2.3 Metric gauge linear N-connections of second order
12.2.4 Einstein – Yang-Mills equations on OscM
12.3 Further extensions of the generalized gauge framework
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 of notions
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.

Los autores
BALAN Vladimir
Full professor at the University «Politehnica» of Bucharest, Romania. Author of scienti?c publications and monographs. Founding member and vice-president of the Balkan Society of Geometers, member of the American Mathematical Society, European Mathematical Society and Romanian Society of Mathematical Sciences. Research areas: Finsler, Lagrange and generalized Lagrange spaces; generalized gauge theory; harmonic maps; variational problems applied to gravitation and relativity, spectral theory of tensors.
RAHULA Мaido
Doctor of Science, professor emeritus of the University of Tartu. As a pedagogue worked in Tartu, Odessa, Algiers. Has carried out investigations in differential geometry, author of scienti?c publications and monographs, member emeritus of the American Mathematical Society, member of honor of the Balkan Society of Geometers, member of honor of the Estonian Mathematical Society.
VOICU Niсoleta
Lecturer. Works at the «Transilvania» University of Brasov, Romania. Her research themes varied over time and include higher order geometries (Miron—Atanasiu theory), topics of pure Finsler geometry and its generalizations, Finsler-based extensions of general relativity, modern variational calculus.