DOCSLIB.ORG

Lectures on Connection Geometry

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lectures on Connection Geometry DGS PPR Lectures on Connection Geometry Phillip E Parker notes by J Ryan Mathematics Department Wichita State University Wichita KS USA philmathwichitaedu ryanmathwichitaedu DRAFT July MSC Primary C Secondary C C Preface This volume is the sequel to and covers the seminar from It assumes a certain familiarity with its predecessor esp ecially Chapter The Tangent Bundle Now we b egin to lo ok at additional structures that can b e imp osed on a manifold to provide it with geometry They all involve some sort of connection or generalized parallel translation so we b egin with the most general kind of connection as conceived by Ehresmann A I wish to thank J Ryan for transcribing my lectures into L T X I also wish E to thank J Ryan and S Sahraei for help with indexing and for comments suggestions and clarications iii Contents Preface iii Standard Symbols vii Preliminaries Algebra Analysis Topology Connections Kinds of connections Horizontal lifts Holonomy Covariant derivative Curvature and holonomy Parallelism and geo desics Automorphisms Exp onential Maps Denition and construction The APS corresp ondence Jacobi elds Gconnections Induced connections Linear Connections Automorphisms Bibliography v vi Contents Index Standard Symbols Number sets and friend N natural Z integer Q rational R real C complex T torus S Categories Bdl bundles cgH kspaces Grp groups Hsdf Hausdor spaces LAlg Lie algebras LGrp Lie groups sometimes their germs Md manifolds Mo d mo dules Rng rings Set sets Vec nitedimensional vector spaces usually real vii Preliminaries We b egin by reviewing some relevant concepts and results from analysis algebra and top ology Algebra Analysis Topology In a top ological category a pair of maps f g X Y is said to admit a homotopy H from f to g if and only if there is a map H X I Y x t H x H x t t H with H f and H g Then we write f g or just f g and say that f and g are homotopic We can also think of H as either a parameter family of maps fH X Y j t g with H f and H g t a curve c from f to g in the space C X Y of maps from X to Y H c C X Y t H H t We call f nul lhomotopic or inessential if it is homotopic to a constant map Intuitively we may picture H as a continuous deformation of the graph of f into that of g The next result is obvious Prop osition Homotopy is an equivalence relation on the set of maps from X to Y Chapter Preliminaries Maps in the same equivalence class of are said to b e homotopic Denition Two top ological spaces X Y are said to b e of the same fheqg homotopy type or homotopy equivalent if and only if there exist continuous maps f X Y and g Y X with g f and f g Then we write X Y X Y and say that f and g are mutual homotopy inverses or inverse up to homotopy Similarly to the case for maps is an equivalence relation on any collection of top ological spaces and one sometimes sp eaks lo osely of spaces in the same class as b eing homotopic Denition A space in the homotopy equivalence class determined by fhptg a singleton space is called contractible or a homotopy point Theorem A ber bund le over a contractible base space is topologically ffbctg trivial proof Even so the bundle might not b e a trivial Gbundle b ecause the Gaction n might not b e trivial The frame bundle on R is one example Connections Kinds of connections In mo dern geometry there are various kinds of connections for a given mani fold M with a bundle structure over it For example A general connection on any bre bundle E M is a splitting of TE into the natural vertical bundle and a horizontal bundle If the splitting is equivariant for the structure group or more generally some subgroup G then it denes an Ehresmann Gconnection A principal connection is an Ehresmann Gconnection on a principal Gbundle P M G A linear connection on a vector bundle E M V GLV over M with mo del b er V is asso ciated to a principal connection on the frame bundle with group GLV All others are nonlinear among which are the ane connections with G A It is unfortunate that in the extant n literature on nonlinear connections for example all written well after a nonlinear connection is dened to b e a particular highly restricted type of connection on TM A Koszul connection is a linear op erator of the type of a covariant deriva tive on a vector bundle It gives rise to a linear connection on the vector bundle A Cartan connection may b e considered as a version of the general con cept of a principal connection in which the geometry of the principal bundle is tied to the geometry of the base manifold Cartan con nections describ e the geometry of manifolds mo delled on homogeneous spaces Under certain technical conditions they can b e related to the remaining types Chapter Connections We shall consider all except Cartan connections We are mostly concerned with nitedimensional real vector bundles E vector spaces V so GLV GLn R GL with n dim V Moreover our main concern is when E n TM so the principal bundle is LM the bundle of linear frames n dim M and linear connections are Gconnections for a suitable subgroup G GL n All pseudoRiemannian connections are connections of this last type Since the fundamental work of Ehresmann we have had a consistent terminology for connections on a manifold M A connection on M is a split ting TTM V H where V is the natural vertical bundle and H is a complementary subbundle the horizontal bundle Our main ob jective is to study smo oth general connections on the tangent bundle TM of a smo oth paracompact connected manifold M We shall use nonlinear in the original sense of Ehresmann We b egin more generally Denition A general connection on a b er bundle F E M is a fcong splitting TE V E H E or just V H for short Since V is natural one frequently refers to H as the general connection The subbundle H is called the horizontal bund le We usually omit the general and refer simply to a connection When E TM it is customary to say the connection is on M To justify calling H the horizontal bundle one observes that for v E p H T M is a vector space isomorphism This follows from V ker v p Denition A connection H is at if and only if H is integrable ffcg In other words if and only if the horizontal spaces foliate TE Recall that the vertical spaces always foliate Denition A connection H on E is trivial if and only if E is a trivial ftcg bundle and the horizontal sections are the constant sections Example The canonical or standard connection on M F is trivial ftcxg Example As the tangent bundle to any Lie group G is trivial the ftcxg canonical or standard connection on G is trivial This gives a canonical at n connection on R that has straight lines as geo desics no matter what signature any co existing inner pro duct might have ftcxg Ex Trivial connections are at Kinds of connections Denition If E is a Gbundle then H is called a Gconnection if fgcg and only if H g H A Gconnection on a principal Gbundle is called a g v v principal Gconnection Denition Let E b e a vector bundle with b erdimension k Then a flcg GL connection on E is called a linear connection k Ex In terms of the horizontal bundle H is a linear connection if and flcg only if H a H for all v E As is customary here we have written a av v as an abbreviation for the induced tangent map of m scalar multiplication a by a Hint use Ex in n Recall the Grassmannian G R G n of k planes in real nspace k k Here we are interested in the splitting TE V H where the b erdimension of TE is k n and the b erdimension of V is k so that the b erdimension of H is n Thus we shall consider Grassmannians of the form G k n n We wish to determine which elements of G k n are admissible as hor n izontal spaces Clearly not all candidates are acceptable such as any element that is not complementary to V Thus we dene the subspace of horizontal v Grassmannians to b e the space of all acceptable candidates Denition The set of all nplanes in G T E complementary to n v fhgg V is called the horizontal Grassmannian and is denoted by G T E v H v G T E n v Note that since V is natural it is xed in G k n Thus G T E H v k G k n is an op en submanifold of G k n Indeed V H fg H n v v The frame bundle LE of TE over E is a principal GL bundle for the k n k n dening action of GL on R As in Ex this induces the k n standard action of GL on G k n and G k n n k n k Denition The asso ciated bundle LE G k n G TE is called h h fhgbdlg the Grassmann hplane bund le over E and G TE LE G k n is the H H horizontal Grassmann bund le over E Observe that a connection H on E may b e regarded as a section of G TE H over E so as H G TE H We shall now determine the subgroup A GL that xes the vertical H k n k space R and acts transitively on horizontal spaces First we may apply any automorphisms of V and H separately Second we may add vertical Chapter Connections comp onents to horizontal vectors to obtain a new horizontal space In blo ck matrix form we write I GL k M I GL n nk
Load more

Recommended publications
  • Connections on Bundles Md
    Dhaka Univ. J. Sci. 60(2): 191-195, 2012 (July) Connections on Bundles Md. Showkat Ali, Md. Mirazul Islam, Farzana Nasrin, Md. Abu Hanif Sarkar and Tanzia Zerin Khan Department of Mathematics, University of Dhaka, Dhaka 1000, Bangladesh, Email: [email protected] Received on 25. 05. 2011.Accepted for Publication on 15. 12. 2011 Abstract This paper is a survey of the basic theory of connection on bundles. A connection on tangent bundle , is called an affine connection on an -dimensional smooth manifold . By the general discussion of affine connection on vector bundles that necessarily exists on which is compatible with tensors. I. Introduction = < , > (2) In order to differentiate sections of a vector bundle [5] or where <, > represents the pairing between and ∗. vector fields on a manifold we need to introduce a Then is a section of , called the absolute differential structure called the connection on a vector bundle. For quotient or the covariant derivative of the section along . example, an affine connection is a structure attached to a differentiable manifold so that we can differentiate its Theorem 1. A connection always exists on a vector bundle. tensor fields. We first introduce the general theorem of Proof. Choose a coordinate covering { }∈ of . Since connections on vector bundles. Then we study the tangent vector bundles are trivial locally, we may assume that there is bundle. is a -dimensional vector bundle determine local frame field for any . By the local structure of intrinsically by the differentiable structure [8] of an - connections, we need only construct a × matrix on dimensional smooth manifold . each such that the matrices satisfy II.
    [Show full text]
  • Proper Affine Actions and Geodesic Flows of Hyperbolic Surfaces
    ANNALS OF MATHEMATICS Proper affine actions and geodesic flows of hyperbolic surfaces By William M. Goldman, Franc¸ois Labourie, and Gregory Margulis SECOND SERIES, VOL. 170, NO. 3 November, 2009 anmaah Annals of Mathematics, 170 (2009), 1051–1083 Proper affine actions and geodesic flows of hyperbolic surfaces By WILLIAM M. GOLDMAN, FRANÇOIS LABOURIE, and GREGORY MARGULIS Abstract 2 Let 0 O.2; 1/ be a Schottky group, and let † H =0 be the corresponding D hyperbolic surface. Let Ꮿ.†/ denote the space of unit length geodesic currents 1 on †. The cohomology group H .0; V/ parametrizes equivalence classes of affine deformations u of 0 acting on an irreducible representation V of O.2; 1/. We 1 define a continuous biaffine map ‰ Ꮿ.†/ H .0; V/ R which is linear on 1 W ! the vector space H .0; V/. An affine deformation u acts properly if and only if ‰.; Œu/ 0 for all Ꮿ.†/. Consequently the set of proper affine actions ¤ 2 whose linear part is a Schottky group identifies with a bundle of open convex cones 1 in H .0; V/ over the Fricke-Teichmüller space of †. Introduction 1. Hyperbolic geometry 2. Affine geometry 3. Flat bundles associated to affine deformations 4. Sections and subbundles 5. Proper -actions and proper R-actions 6. Labourie’s diffusion of Margulis’s invariant 7. Nonproper deformations 8. Proper deformations References Goldman gratefully acknowledges partial support from National Science Foundation grants DMS- 0103889, DMS-0405605, DMS-070781, the Mathematical Sciences Research Institute and the Oswald Veblen Fund at the Insitute for Advanced Study, and a Semester Research Award from the General Research Board of the University of Maryland.
    [Show full text]
  • Elementary Differential Geometry
    ELEMENTARY DIFFERENTIAL GEOMETRY YONG-GEUN OH { Based on the lecture note of Math 621-2020 in POSTECH { Contents Part 1. Riemannian Geometry 2 1. Parallelism and Ehresman connection 2 2. Affine connections on vector bundles 4 2.1. Local expression of covariant derivatives 6 2.2. Affine connection recovers Ehresmann connection 7 2.3. Curvature 9 2.4. Metrics and Euclidean connections 9 3. Riemannian metrics and Levi-Civita connection 10 3.1. Examples of Riemannian manifolds 12 3.2. Covariant derivative along the curve 13 4. Riemann curvature tensor 15 5. Raising and lowering indices and contractions 17 6. Geodesics and exponential maps 19 7. First variation of arc-length 22 8. Geodesic normal coordinates and geodesic balls 25 9. Hopf-Rinow Theorem 31 10. Classification of constant curvature surfaces 33 11. Second variation of energy 34 Part 2. Symplectic Geometry 39 12. Geometry of cotangent bundles 39 13. Poisson manifolds and Schouten-Nijenhuis bracket 42 13.1. Poisson tensor and Jacobi identity 43 13.2. Lie-Poisson space 44 14. Symplectic forms and the Jacobi identity 45 15. Proof of Darboux' Theorem 47 15.1. Symplectic linear algebra 47 15.2. Moser's deformation method 48 16. Hamiltonian vector fields and diffeomorhpisms 50 17. Autonomous Hamiltonians and conservation law 53 18. Completely integrable systems and action-angle variables 55 18.1. Construction of angle coordinates 56 18.2. Construction of action coordinates 57 18.3. Underlying geometry of the Hamilton-Jacobi method 61 19. Lie groups and Lie algebras 62 1 2 YONG-GEUN OH 20. Group actions and adjoint representations 67 21.
    [Show full text]
  • Geometric Control of Mechanical Systems Modeling, Analysis, and Design for Simple Mechanical Control Systems
    Francesco Bullo and Andrew D. Lewis Geometric Control of Mechanical Systems Modeling, Analysis, and Design for Simple Mechanical Control Systems – Supplementary Material – August 1, 2014 Contents S1 Tangent and cotangent bundle geometry ................. S1 S1.1 Some things Hamiltonian.......................... ..... S1 S1.1.1 Differential forms............................. .. S1 S1.1.2 Symplectic manifolds .......................... S5 S1.1.3 Hamiltonian vector fields ....................... S6 S1.2 Tangent and cotangent lifts of vector fields .......... ..... S7 S1.2.1 More about the tangent lift ...................... S7 S1.2.2 The cotangent lift of a vector field ................ S8 S1.2.3 Joint properties of the tangent and cotangent lift . S9 S1.2.4 The cotangent lift of the vertical lift ............ S11 S1.2.5 The canonical involution of TTQ ................. S12 S1.2.6 The canonical endomorphism of the tangent bundle . S13 S1.3 Ehresmann connections induced by an affine connection . S13 S1.3.1 Motivating remarks............................ S13 S1.3.2 More about vector and fiber bundles .............. S15 S1.3.3 Ehresmann connections ......................... S17 S1.3.4 Linear connections and linear vector fields on vector bundles ....................................... S18 S1.3.5 The Ehresmann connection on πTM : TM M associated with a second-order vector field→ on TM . S20 S1.3.6 The Ehresmann connection on πTQ : TQ Q associated with an affine connection on Q→.......... S21 ∗ S1.3.7 The Ehresmann connection on πT∗Q : T Q Q associated with an affine connection on Q ..........→ S23 S1.3.8 The Ehresmann connection on πTTQ : TTQ TQ associated with an affine connection on Q ..........→ S23 ∗ S1.3.9 The Ehresmann connection on πT∗TQ : T TQ TQ associated with an affine connection on Q ..........→ S25 ∗ S1.3.10 Representations of ST and ST ..................
    [Show full text]
  • Complete Connections on Fiber Bundles
    Complete connections on fiber bundles Matias del Hoyo IMPA, Rio de Janeiro, Brazil. Abstract Every smooth fiber bundle admits a complete (Ehresmann) connection. This result appears in several references, with a proof on which we have found a gap, that does not seem possible to remedy. In this note we provide a definite proof for this fact, explain the problem with the previous one, and illustrate with examples. We also establish a version of the theorem involving Riemannian submersions. 1 Introduction: A rather tricky exercise An (Ehresmann) connection on a submersion p : E → B is a smooth distribution H ⊂ T E that is complementary to the kernel of the differential, namely T E = H ⊕ ker dp. The distributions H and ker dp are called horizontal and vertical, respectively, and a curve on E is called horizontal (resp. vertical) if its speed only takes values in H (resp. ker dp). Every submersion admits a connection: we can take for instance a Riemannian metric ηE on E and set H as the distribution orthogonal to the fibers. Given p : E → B a submersion and H ⊂ T E a connection, a smooth curve γ : I → B, t0 ∈ I, locally defines a horizontal lift γ˜e : J → E, t0 ∈ J ⊂ I,γ ˜e(t0)= e, for e an arbitrary point in the fiber. This lift is unique if we require J to be maximal, and depends smoothly on e. The connection H is said to be complete if for every γ its horizontal lifts can be defined in the whole domain. In that case, a curve γ induces diffeomorphisms between the fibers by parallel transport.
    [Show full text]
  • Smoothing Maps Into Algebraic Sets and Spaces of Flat Connections
    SMOOTHING MAPS INTO ALGEBRAIC SETS AND SPACES OF FLAT CONNECTIONS THOMAS BAIRD AND DANIEL A. RAMRAS n Abstract. Let X ⊂ R be a real algebraic set and M a smooth, closed manifold. We show that all continuous maps M ! X are homotopic (in X) to C1 maps. We apply this result to study characteristic classes of vector bundles associated to continuous families of complex group representations, and we establish lower bounds on the ranks of the homotopy groups of spaces of flat connections over aspherical manifolds. 1. Introduction The first goal of this paper is to prove the following result about the differential topology of algebraic sets. Theorem 1.1 (Section2) . Let X ⊂ Rn be a (possibly singular) real algebraic set, and let f : M ! X be a continuous map from a smooth, closed manifold M. Then there exists a map g : M ! X, and a homotopy H : M × I ! X connecting f and g g, such that the composite M ! X,! Rn is C1. The problem of smoothing maps into algebraic sets seems natural, but we have not found mention of it in the literature. We consulted several experts in real algebraic geometry; some expected our result to hold, and some did not. Our proof proceeds by embedding X as the singular set of an irreducible, quasi- projective variety Y and using a resolution of singularities Ye ! Y for which the inverse image of X is a divisor with normal crossing singularities. Basic facts about neighborhoods of algebraic sets then reduce the problem to the case of normal crossing divisors, which can be handled by differential-geometric means.
    [Show full text]
  • WHAT IS a CONNECTION, and WHAT IS IT GOOD FOR? Contents 1. Introduction 2 2. the Search for a Good Directional Derivative 3 3. F
    WHAT IS A CONNECTION, AND WHAT IS IT GOOD FOR? TIMOTHY E. GOLDBERG Abstract. In the study of differentiable manifolds, there are several different objects that go by the name of \connection". I will describe some of these objects, and show how they are related to each other. The motivation for many notions of a connection is the search for a sufficiently nice directional derivative, and this will be my starting point as well. The story will by necessity include many supporting characters from differential geometry, all of whom will receive a brief but hopefully sufficient introduction. I apologize for my ungrammatical title. Contents 1. Introduction 2 2. The search for a good directional derivative 3 3. Fiber bundles and Ehresmann connections 7 4. A quick word about curvature 10 5. Principal bundles and principal bundle connections 11 6. Associated bundles 14 7. Vector bundles and Koszul connections 15 8. The tangent bundle 18 References 19 Date: 26 March 2008. 1 1. Introduction In the study of differentiable manifolds, there are several different objects that go by the name of \connection", and this has been confusing me for some time now. One solution to this dilemma was to promise myself that I would some day present a talk about connections in the Olivetti Club at Cornell University. That day has come, and this document contains my notes for this talk. In the interests of brevity, I do not include too many technical details, and instead refer the reader to some lovely references. My main references were [2], [4], and [5].
    [Show full text]
  • Automorphisms of Ehresmann Connections
    Acta Math. Hungar., 123 (4) (2009), 379395. DOI: 10.1007/s10474-008-8139-x First published online December 12, 2008 AUTOMORPHISMS OF EHRESMANN CONNECTIONS J. PÉK and J. SZILASI¤ Institute of Mathematics, University of Debrecen, H-4010 Debrecen, P.O.B. 12, Hungary e-mail: [email protected] (Received July 14, 2008; accepted July 28, 2008) Abstract. We prove that a dieomorphism of a manifold with an Ehresmann connection is an automorphism of the Ehresmann connection, if and only if, it is a totally geodesic map (i.e., sends the geodesics, considered as parametrized curves, to geodesics) and preserves the strong torsion of the Ehresmann connection. This result generalizes and to some extent strengthens the classical theorem on the automorphisms of a D-manifold (manifold with covariant derivative). 1. Introduction It is well-known (and almost trivial) that two covariant derivative oper- ators on a manifold are equal, if and only if, they have the same geodesics and equal torsion tensors. Heuristically, this implies that a dieomorphism of a manifold with a covariant derivative is an automorphism of the covari- ant derivative, if and only if, it is a totally geodesic map (i.e. sends geodesics, considered as parametrized curves, to geodesics) and preserves the torsion. ¤The second author is supported by the Hungarian Nat. Sci. Found. (OTKA), Grant No. NK68040. Key words and phrases: Ehresmann connection, automorphism, strong torsion, totally geodesic map. 2000 Mathematics Subject Classication: 53C05, 53C22. 02365294/$ 20.00 °c 2008 Akadémiai Kiadó, Budapest 380 J. PÉK and J. SZILASI A formal proof of this conceptually important result may be found e.g.
    [Show full text]
  • Principal Bundles, Vector Bundles and Connections
    Appendix A Principal Bundles, Vector Bundles and Connections Abstract The appendix defines fiber bundles, principal bundles and their associate vector bundles, recall the definitions of frame bundles, the orthonormal frame bun- dle, jet bundles, product bundles and the Whitney sums of bundles. Next, equivalent definitions of connections in principal bundles and in their associate vector bundles are presented and it is shown how these concepts are related to the concept of a covariant derivative in the base manifold of the bundle. Also, the concept of exterior covariant derivatives (crucial for the formulation of gauge theories) and the meaning of a curvature and torsion of a linear connection in a manifold is recalled. The concept of covariant derivative in vector bundles is also analyzed in details in a way which, in particular, is necessary for the presentation of the theory in Chap. 12. Propositions are in general presented without proofs, which can be found, e.g., in Choquet-Bruhat et al. (Analysis, Manifolds and Physics. North-Holland, Amsterdam, 1982), Frankel (The Geometry of Physics. Cambridge University Press, Cambridge, 1997), Kobayashi and Nomizu (Foundations of Differential Geometry. Interscience Publishers, New York, 1963), Naber (Topology, Geometry and Gauge Fields. Interactions. Applied Mathematical Sciences. Springer, New York, 2000), Nash and Sen (Topology and Geometry for Physicists. Academic, London, 1983), Nicolescu (Notes on Seiberg-Witten Theory. Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2000), Osborn (Vector Bundles. Academic, New York, 1982), and Palais (The Geometrization of Physics. Lecture Notes from a Course at the National Tsing Hua University, Hsinchu, 1981). A.1 Fiber Bundles Definition A.1 A fiber bundle over M with Lie group G will be denoted by E D .E; M;;G; F/.
    [Show full text]
  • Projective Connections and the Algebra of Densities
    , Projective Connections and the Algebra of Densities Jacob George School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK Abstract. Projective connections first appeared in Cartan’s papers in the 1920’s. Since then they have resurfaced periodically in, for example, integrable systems and perhaps most recently in the context of so called projectively equivariant quantisation. We recall the notion of projective connection and describe its relation with the algebra of densities on a manifold. In particular, we construct a Laplace-type operator on functions using a Thomas projective connection and a symmetric contravariant tensor of rank 2 (‘upper metric’). Keywords: Projective connections, densities PACS: PACS 02.40.Dr Euclidean and projective geometries, PACS 02.10.De Algebraic structures and number theory INTRODUCTION This paper is concerned with the geometry of differential operators on a manifold and their relation with projective connections. The notion of projective connection is an old one, first appearing in Cartan’s papers of the 1920s and then in various modified forms throughout the 20th century. They have surfaced periodically in mathematical physics, in particular in integrable systems and more recently in projectively equivariant quantisation. This paper establishes that the algebra of densities introduced in [1] and projective connections on a given manifold are fundamentally linked. We recall the notion of a projective connection; in fact there are two distinct but related notions which will be detailed here. We define the manifold M (cf. Mˆ in [1]) for which the algebra of densities V(M) may be interpreted as a subalgebra of the algebra of smooth ∞ b functions C (M).
    [Show full text]
  • The Connections of Pseudo-Finsler Spaces
    The connections of pseudo-Finsler spaces E. Minguzzi∗ Abstract We give an introduction to (pseudo-)Finsler geometry and its connections. For most results we provide short and self contained proofs. Our study of the Berwald non-linear connection is framed into the theory of connec- tions over general fibered spaces pioneered by Mangiarotti, Modugno and other scholars. The main identities for the linear Finsler connection are presented in the general case, and then specialized to some notable cases like Berwald’s, Cartan’s or Chern-Rund’s. In this way it becomes easy to compare them and see the advantages of one connection over the other. Since we introduce two soldering forms we are able to characterize the notable Finsler connections in terms of their torsion properties. As an application, the curvature symmetries implied by the compatibility with a metric suggest that in Finslerian generalizations of general relativity the mean Cartan torsion vanishes. This observation allows us to obtain dy- namical equations which imply a satisfactory conservation law. The work ends with a discussion of yet another Finsler connection which has some advantages over Cartan’s and Chern-Rund’s. Dedicated to Marco Modugno Contents 1 Introduction 2 2 Geometrical preliminaries 3 arXiv:1405.0645v2 [math-ph] 1 Aug 2014 2.1 Vectorbundleandlocalcoordinates . 3 2.2 Vertical space and Liouville vector field . 4 2.3 Thetangentbundle.......................... 5 2.4 Secondorderequations........................ 6 3 The non-linear connection 7 3.1 Connectionsonvectorbundles . 7 3.2 Covariantderivatives......................... 9 3.3 Curvature and torsion of the non-linear connection . 11 3.4 Relationship between non-linear connections and sprays .
    [Show full text]
  • An Brief Introduction to Finsler Geometry
    An brief introduction to Finsler geometry Matias Dahl July 12, 2006 Abstract This work contains a short introduction to Finsler geometry. Special em- phasis is put on the Legendre transformation that connects Finsler geometry with symplectic geometry. Contents 1 Finsler geometry 3 1.1 Minkowski norms . 4 1.2 Legendre transformation . 6 2 Finsler geometry 9 2.1 The global Legendre transforms . 9 3 Geodesics 15 4 Horizontal and vertical decompositions 19 4.1 Applications . 22 5 Finsler connections 25 5.1 Finsler connections . 25 5.2 Covariant derivative . 26 5.3 Some properties of basic Finsler quantities . 27 6 Curvature 29 7 Symplectic geometry 31 7.1 Symplectic structure on T ∗M n f0g . 33 7.2 Symplectic structure on T M n f0g . 34 1 Table of symbols in Finsler geometry For easy reference, the below table lists the basic symbols in Finsler ge- ometry, their definitions (when short enough to list), and their homogene- ity (see p. 3). However, it should be pointed out that the quantities and notation in Finsler geometry is far from standardized. Some references (e.g. [Run59]) work with normalized quantities. The present work follows [She01b, She01a]. Name Notation Homogeneity (co-)Finsler norm F; H; h 1 1 @2F 2 gij = 2 @yi @yj 0 ij g = inverse of gij 0 1 @gij 1 @3F 2 Cartan tensor Cijk = 2 @yk = 4 @yi @yj @yk −1 i is Cjk = g Csjk −1 @Cijk Cijkl = @yl −2 Geodesic coefficients Gi 2 G i @ i @ Geodesic spray = y @xi − 2G @yi no i @Gi Non-linear connection Nj = @yj 1 δ @ s @ Horizontal basis vectors δxi = @xi − Ni @ys no i i @Nj @2Gi Berwald connection Gjk = @yk = @yj @yk 0 Chern-Rund connection Γijk 0 i is Γjk = g Γsjk 0 Landsberg coefficients Lijk m Curvature coefficients Rijk 0 m Rij 1 m kL −1 Rij = Rijky m 2 i ij Legendre transformation L = h ξj 1 L ∗ L −1 j : T M ! T M i = gijy 1 2 1 Finsler geometry Essentially, a Finsler manifold is a manifold M where each tangent space is equipped with a Minkowski norm, that is, a norm that is not necessarily induced by an inner product.
    [Show full text]