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Cover Balan V., Rahula M., Voicu N. Tangent Structures in Geometry and Their Applications
Id: 180208

Tangent Structures in Geometry and Their Applications

URSS. 448 pp. (English). Paperback. ISBN 978-5-396-00588-4.
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N Differential prolongations are usually obtained by means of differentiation and jets of mappings which are, in one way or another, related to local coordinates. The present book sets the foundation of prolongation theory on iterated tangent bundles, in a coordinate-free manner. Lie-Cartan calculus, the theory of connections in bundles and certain specific structures of Finsler geometry are developed in an invariant form. Applications of this approach include: electromagnetic field theory, generalized gauge fields, Hamilton, Lagrange, Maxwell and Einstein---Yang---Mills equations, Berwald---Moor connections, Jacobi-type stability problems and KCC-theory.

The book is mainly intended for scientific researchers, but it can be also used as an advanced textbook. To this aim, the text contains numerous exercises and illustrative examples.


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


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.


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.



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


-- 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

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.


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