| 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 Delta_{h} \oplus Delta_{v} |

| | 4.7.1 | Horizontal distribution Delta_{h} |

| | 4.7.2 | Specialized basis |

| | 4.7.3 | Adapted basis |

| | 4.7.4 | Symmetries of the distribution Delta_{h} |

| | 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 F^{alpha}_{ij} |

| | 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 J_{n,m} |

| | 5.2.1 | Multi-indices |

| | 5.2.2 | Multi-dimensional time |

| | 5.2.3 | Embedding in J_{n,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 *T*^{2}M |

| | 8.3.1 | The second tangent bundle *T*^{2}M |

| | 8.3.2 | Lifts of curves and of families of curves to *T*^{2}M |

| | 8.3.3 | A special Ehresmann connection on *T*^{2}M |

| 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 H_{n} |

| | 9.9 | The Berwald-Moór dual structure |

| | 9.10 | Hamilton vs. Lagrange equations |

| | 9.11 | Geometric structures on T*H_{n} |

| 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 Osc*M* |

| | 12.2.1 | Gauge transformations on Osc*M* |

| | 12.2.2 | Gauge covariant derivatives on Osc*M* |

| | 12.2.3 | Metric gauge linear *N*-connections of second order |

| | 12.2.4 | Einstein - Yang-Mills equations on Osc*M* |

| | 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 |

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.

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.

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.