a thesis about connections

A HISTORICAL OVERVIEW OF CONNECTIONS IN GEOMETRY A Thesis by Kamielle Freeman Bachelor of Arts, Wichita State University, 2008 Submitted to the Department of Mathematics and Statistics and the faculty of the Graduate School of Wichita State University in partial fulllment of the requirements for the degree of Master of Science May 2011 c Copyright 2011 by Kamielle Freeman All Rights Reserved A HISTORICAL OVERVIEW OF CONNECTIONS IN GEOMETRY The following faculty members have examined the nal copy of this thesis for form and content, and recommend that it be accepted in partial fulllment of the requirement for the degree of Master of Science with a major in Mathematics. Phillip E. Parker, Committee Chair Thalia Jeres, Committee Member Elizabeth Behrman, Committee Member iii DEDICATION To Pandora. This thesis was accomplished for you and in spite of you. iv \Mathematics is very akin to Art; a mathematical theory not only must be rigorous, but it must also satisfy our mind in quest of simplicity, of harmony, of beauty..." |Charles Ehresmann v ACKNOWLEDGEMENTS I would like to thank Esau Freeman and Denise Unruh for making this thesis possible through their loving support, Dr. Phillip E. Parker for his mentoring and excellent mathe- matical instruction, Emma Traore Zohner for help translating, Raja Balakrishnan and Justin Ryan for their helpful conversations and proof-reading, and all of my friends on the 3rd oor of Jabara Hall for their encouraging words and moral support. vi ABSTRACT This thesis is an attempt to untangle/clarify the modern theory of connections in Geometry. Towards this end a historical approach was taken and original as well as secondary sources were used. An overview of the most important historical developments is given as well as a modern look at how the various denitions of connection are related. vii PREFACE In this thesis I hope to clear up some of the confusions surrounding a connection in Dierential Geometry; or, at least some of the things that confused me when I was trying to gure out just what exactly a connection is. In particular, 1) How is a covariant derivative operator related to a connection? 2)How is parallel transport related to the previous two notions? 3) What was the rst denition of a connection? 4) What is a connection in the most general sense? I hope that I answer all of these questions satisfactorily in the following pages. I have also provided a chart of the heirarchy of connections via the use of Lie subgroups. viii TABLE OF CONTENTS Chapter Page 1 THE BACK STORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 \Anti"- Euclidean Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The Dierential Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 FIRST ENCOUNTERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2r. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3P. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 A First Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 GENERALIZATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.1 Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Fiber Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3 A Minimal Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4 MODERNITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.1 Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.2 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 A Note on ane connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 B Note on Cartan connections . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 C Chart of connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 ix CHAPTER 1 THE BACK STORY The motivation for the idea of \connection" comes from two separate developments in the history of Mathematics; namely, those of Non-Euclidean Geometry and Dierential Calculus. The concept of \connection" is crucial to both. 1.1 \Anti"- Euclidean Space Ever since Euclid conceived it, the parallel postulate has been subject to much con- troversy. For a long time it was believed to be provable by the other four postulates. In fact several attempts at proving the parallel postulate from the other four were actually accepted as proofs, but eventually a mistake was found. By 1817 Carl Friedrich Gauss had become convinced that the fth postulate was independent of the other four postulates and that a consistent geometry, dierent from the Euclidean one, was possible. At that time he referred to such a geometry as \anti-Euclidean". In 1824 the term \non-Euclidean Geometry" ap- peared for the rst time in a letter Gauss sent to Franz Adolf Taurinus who was one of the rst to publish results about non-Euclidean Geometry.[6] At the time Gauss was working on this, thinking was dominated by the philosophy of Immanuel Kant who had stated that Euclidean geometry is the inevitable necessity of thought . It is said that Gauss disliked con- troversy and thus kept his \anti-Euclidean" work to himself. Despite Gauss' reticence, the idea of a non-Euclidean geometry gained ground over the next 3 decades thanks especially to the publications of J anos Bolyai and Nikolai Lobachevsky. However, it would be a long while yet before this new Geometry was unconditionally accepted by pure Geometers. In [11] K.Reich suggests that the missing concept of parallelism may have been partially to blame for the delay, as it has always been a central theme for the foundations of Geometry. When Gauss developed the theory of surfaces during the years 1825-1827 he used the fundamental property that \there exist parametrizations of any surface by coordinate 1 systems which produce vectors in the tangent plane at that point." Gauss focused mainly on the in uence of curvature on the geometry of the surface using geodesic curves on surfaces embedded in 3-dimensional Euclidean space. In doing so, he established geometric properties extrinsically and then made them intrinsic by proving that the properties really do not depend on the embedding. His was the rst systematic use of geodesics to make a general Geometry, yet it did not include a notion of parallelism.[6] The generalization to higher dimensions of what Gauss did for surfaces is due to Bern- hard Riemann and is what makes up the content of his lecture Uber die Hypothesen welche der Geometrie zu Grunde liegen (\On the hypotheses that lie at the foundation of Geom- etry"). In Part I of Riemann's lecture, the concept of an \ n-fold extended quantity" (now called a manifold of dimension n) is introduced. Although the details are fuzzy (there are no equations in this particular work of Riemann's as it was intended for non-mathematicians), it is clear that Riemann's \ n-dimensional spaces" are characterized by the fact that they are locally like n-dimensional Euclidean space, which Riemann referred to as at.[6] In Part II Riemann considers what are now called Riemannian metrics and curvature. He uses, just as Gauss did, the shortest path to build a geodesic coordinate system. Although Dedekind in his biography of Riemann called the lecture \a masterpeice of exposition", it is generally believed that Gauss was the only attendee who really appreciated what Riemann had done. Not surprising that Gauss should be so impressed as he was Riemann's advisor at the time and even chose the topic. Regardless of its reception at the time, no one today would deny that the ideas contained within are some of the most important in the entire history of Dierential Geometry. Riemann, like Gauss, did not include a concept of parallelism; in fact, the word parallel was not even mentioned.[11] 1.2 The Dierential Calculus The idea of a derivative was considered by several European mathematicians in the early 17th century, but Isaac Newton and Gottfried Leibniz must share the credit (even if they do it grudgingly) for their independent developments of The Calculus (classically 2 referred to as \innitesimal calculus") which consisted of Dierential Calculus and Integral Calculus. When dening the derivative, Isaac Newton considered variables as changing with respect to time; Leibniz thought of variables xandyas ranging over sequences of innitely close values. Both Newton and Leibniz provided a formula for nding the second derivative. In Euclidean space, the second derivative of a dependent variable ywith respect to an independent variable xis represented (in Leibniz's notation, still used today) asd2y dx2and generalizes to higher dimensions via partial derivatives as@2f @x2wheref:Rn!R. With the rise of non-Euclidean geometry there also arose a problem with the Dif- ferential Calculus. To calculate the second derivative it is necessary to compare tangent vectors at distinct points of curves in space. If the space is Euclidean, vectors can be easily compared because as long as the magnitude and direction of a vector stays the same, it can be moved about freely. However, in the case of curves restricted to surfaces this no longer works. Without a specied method of how it was to be done, vectors were restricted to their base points. When Riemann developed his n-dimensional surfaces (now known as dierential manifolds), there still was no prescription for moving vectors based at one point of a space to another point and thus no satisfactory solution to the problem of how to dene second derivatives (i.e., acceleration) on curved surfaces. 3 CHAPTER 2 FIRST ENCOUNTERS The concept of connection was implemented at least as early as 1869 but it was almost a half century later before the mathematical community recognized a connection for what it actually was. 2.1 In 1869 Elwin Bruno Christoel published Uber die Transformation der homoge- nen Dierentialausdr ucke zweiten Grades in which he introduced the famous symbols k ij (Christoel's notation wask ji ). In his paper, Christoel is interested in \conditions that dierential expressions of degree 2 must verify to be able to transform one into another by a change of variables." Toward this end Christoel considers two expressions of the same Riemannian metric ( g) and begins by assuming the relation nX i;j=1gij(xk)dxidxj=nX ;=1g(y )dydy where the coordinates ( xk) and (y ) and their dierentials are related by the transformation f(i.e.,y=f(x)). In order to isolate a second partial derivative of the transformation f, Christoel introduces the coecients noted today k ij=1 2nX l=1gkl@gjl @xi+@gil @xj@gji @xl : Thus, he obtains the fundamental equation @2x @xi@xl=nX j=1j li@f @xjnX =1 @f @xi@f @xl; () which describes the way in which the k ijtransform under change of coordinates.[6] Christoel's point of view is strictly that of a mathematician interested in \invariant theory". The goal of his paper was to solve a problem of \classication" and he certainly 4 did not investigate any relationships with any particular curves in space.[6] What Christoel did not realize was the fact that the symbols he had discovered themselves determine a connection . Specically, they are the \coecients" of the connection. In fact, a generalized form of equation ( ) is used in [13] to dene a ( classical )connection as \an assignment ofn3numbers to each coordinate system, such that equation ( ) holds" (i.e., connection coecients in more general settings). 2.2r It was Gregorio Ricci-Curbastro who noticed that the Christoel symbols obtained from the Riemannian metric could be used to create a \ coordinate free" dierential calculus. This extension of the Dierential Calculus allowed a modication of partial dierentiation to spaces with curvature (i.e. non-Euclidean) by means of a covariant derivative. As stated in [6], Ricci dened the covariant derivative of a vector eld X(X= n j=1Xj@ @xjin local coordinates) to be: A eld of endomorphisms of TM whose components Xi ;jare calculated from the components ( Xi) ofX, by the formula1 Xi ;j=@Xi @xj+nX k=1k ijXk: In Euclidean space, the sum is zero and hence the covariant derivative is just an ordinary partial derivative. Ricci takes great care that an equation derived in one coordinate system is valid in any other. This coordinate independence is the reason for the name \Absolute" which was given to this extension of the Dierential Calculus. In his 1893 publication Ricci explains: \For the sake of brevity I will denote by the name the `absolute dierential calculus' the totality of methods which I named in other occasions covariant and contravariant derivatives, because they can be applied to any fundamental form independently of the choice of the independent variables and require instead that the latter are completely general and arbitrary."[2] 1In modern notation the covariant derivative Xi ;jis written as (rjX)ior (r@jX)ior (r@ @xiX)iwherer is referred to as a covariant derivative operator . 5 Ricci developed his calculus throughout the years 1884-1900 but despite the signi- cance of what Ricci had discovered, the Absolute Dierential Calculus remained relatively unknown. In 1899 Ricci, with the help of his student Tullio Levi-Civita, wrote a nal ex- position on the Absolute Dierential Calculus. It appeared in the Mathematische Annalen in 1901. The goal of this nal publication was to \put the [ADC] within the capacity of everybody who needed it". However, as stated by Levi-Civita himself, \for many more years it was used almost exclusively by its inventor and a few of his students". The essential shift in appreciation for the Absolute Dierential Calculus resulted from Einstein's theory of General Relativity. As Levi-Civita stated, this physical theory was \the great trial, foretold by Beltrami, where Ricci's calculus revealed itself to be not only useful but truly indispens- able". However, even at this point, the new calculus was purely formal without geometric interpretation.[2] At this point a connection is essentially right under the nose of the mathematical community; in the form of a covariant derivative, which stayed purely an analytic algorithm until 1917. 2.3P Tullio Levi-Civita introduced the concept of parallelism on a Riemannian manifold in his paper Nozione di parallelismo in una variet a qualunque e conseguente specicazione geometrica della curvatura riemanniana published in 1917. There were two other indepen- dent discoveries of \parallelism" by G. Hessenberg and J.A. Schouten. However, Levi-Civita had priority of publication and so today it is referred to as \Levi-Civita's parallelism".[11] Levi-Civita began his famous work with the goal of reducing the formal apparatus com- monly used in order to introduce the Riemann symbols relative to the curvature of a certain 4-dimensional manifold and he did make some progress toward this goal. However, it was the geometric interpretation of the Riemann curvature and the notion of parallel directions in an arbitrary Riemannian manifold discovered along the way which makes up the majority of the paper and for which it is so well known today.[7] 6 Levi-Civita's original idea began with the observation that any Riemannian manifold Vncould be embedded in a Euclidean space of higher dimension SN. Although manifolds weren't fully understood until at least sometime in the 1950s, it is indeed what Levi-Civita had in mind. He then used parallel displacement in the embedding space to move a tangent vectoratPto the nearby point P0. The resulting vector is not necessarily in the tangent space atP0; however, by orthogonal projection onto the tangent space at P0Levi-Civita obtains a vector 0which is the tangent vector at P0parallel to the tangent vector atP. This denition is not intrinsic but Levi-Civita saw that if P0is restricted to being a point innitesimally close to P, then the denition can be made intrinsic and the parallel vector is determined by the metric.[6] Indeed, let xi(i= 1;:::;n ) be the general coordinates of Vn,dxithe increments corresponding to the passage from PtoP0,(i)the parameters related to any direction () going out of P, and(i)+d(i)those related to an innitely close direction (0) going out of P0. Then, according to Levi-Civita the parallelism condition is expressed by the nequations d(i)+ n j;i=1n jl io dxj(l)= 0 wheren jl io is the Christoel symbol. These equations are invariant for any transformation of coordinates and contain only quantities given in the manifold Vn. `Once the law is known according to which one passes from a point to a point innitely close to it, one is able immediately to accomplish the displacement of parallel directions along any arbitrary curve C'(Levi-Civita, 1917,p.3). Assuming that xi=xi(s) are the parametric equations of the curve, then the functions(i)(s) are determined by the equations d(i) ds+ n j;l=1n jl iodxj ds(l)= 0: As a consequence of this denition it follows that the parallel displacement from a pointPto another point P0ofVndepends upon the choice of curve joining them 7 along which the parallel displacement is made. In addition, Levi-Civita remarked that the directions of the tangents along the same geodesic are all mutually parallel; this is a generalization to arbitrary manifolds of a basic property of straight lines in Euclidean spaces. He also observed that the parallel displacement along any arbitrary path of two dierent directions preserves the angle formed by them.[2] Levi-Civita is frequently given credit for the discovery and naming of a \connection". What he actually discovered was a parallelism structure P. The signicance of Levi-Civita's 1917 paper was in the geometric information that it gleaned about the covariant deriva- tive. In the special case of a manifold isometrically embedded into a higher dimensional Euclidean space, the covariant derivative can be viewed as the orthonormal projection of the Euclidean derivative along a tangent vector onto the manifold's tangent space. Once Levi- Civita pointed out the parallelism that was inherent in the covariant derivative operator, it was not long before the theory of connections was truly born. 2.4 A First Denition The term \connection" rst appears in Hermann Weyl's 1918 text Reine Innitesimal Geometrie . In section 3 he describes an ane connection as \that which determines into which vector at P0a vector at an innitesimally close point Pwill transform under parallel displacement from PtoP0." A condition which Weyl requires of parallel displacement in this instance is that the transfer of the totality of vectors from Pto the innitely close point P0by means of parallel displacement produces an ane transformation of the vectors at Pto the vectors at P0. The transformation is an ane one in the sense that it preserves collinearity and ratios of distances but not necessarily angles or lengths. Weyl goes on to dene components of the ane connection to be quantities i rs= i sr. In section 4 Weyl denes a metric connection to be an ane connection on a metric manifold (by which he must mean a Riemannian manifold since the associated quadratic form is now assumed to be positive denite; i.e. a Riemannian metric) where the i rsare Christoel symbols.[14] 8 Levi-Civita's parallel displacement via a covariant derivative operation certainly ts Weyl's denition of \connection" and is why today Levi-Civita's system of parallel transport Pis called the Levi-Civita connection , as is the covariant derivative operator rassociated to it. At this point the reader should recognize that the concept of connection has indeed xed two important problems. A connection, in the form of a covariant derivative, allows for a well dened second derivative on a curved surface in any pseudo-Riemannian space (and thus, Riemannian and Euclidean spaces as well). A connection, in the form of a system of parallel transport, allows for a notion of parallelism in the same. 9 CHAPTER 3 GENERALIZATION Directly after the introduction of the theory of connections in 1917, a generalization for spaces other than Riemannian, and situations other than those specic to the Levi- Civita connection , was sought. Weyl was the rst to make a successful generalization. His \ane connection" successfully generalized the \Levi-Civita connection" for spaces other than Riemannian. Weyl's was only a small step in the process of generalization. During the next couple of decades E. Cartan took center stage in the development of connection theory as he saught to bring F. Klein's view of Geometry to bear upon Dierential Geometry. By 1950 this view was subsumed to an even wider frame; that of connections in principal ber bundles. 3.1 Lie groups Into the burgeoning eld of Dierential Geometry came the algebraic concept of a group. A Lie group is essentially a group which is also a dierentiable manifold so that the group operations are compatible with the smooth structure. Such groups are named for Sophus Lie who laid the foundations with his work on continuous transformation groups. At the time Lie was working on this, his friend and colleague Felix Klein was working on what is today called the Erlangen Program (because the work took place in Erlangen, Germany). In 1872 Klein published his in uential research program under the title Vergleichende Betra- chtungen uber neuere geometrische Forschungen . Klein's Erlangen Program was innovative in several ways, one of which was the idea that group theory was the most useful way of organizing geometrical knowledge. By this time, several non-Euclidean geometries had al- ready emerged, but their relationships with one another had not been adequately claried. Klein's program showed that a geometry could be dened by the properties of a space that are invariant under a given group of transformations. However, Klein's view of Geometry 10 applied to whole spaces. \Localizing it, as one might say, and attaining clarity about what was part of the denition and what had to be added on (or might even be incompatible) was by no means an obvious move to make."[6] By the time Einstein developed the Theory of General Relativity, Elie Cartan was already an expert in the theory of innitesimal Lie groups (i.e., Lie algebras ) which he had been working on at least since 1892. During that year Lie came to Paris at the invitation of Darboux. The main purpose of his visit was to meet Cartan and the two had many discussions during Lie's six month stay. In one article Cartan writes that Lie was interested \with a great good will in the research of young French mathematicians" and that \Posterity will see in him only the genius who created the theory of transformation groups, and we French shall never be able to forget the ties, which bind us to him and which make his memory dear to us."[1] Once he had come into contact with Einstein's theory, Cartan immediately began working on a general framework which would link Dierential Geometry to an \innites- imalized generalization of Klein's Erlangen program."[12] Toward this end Cartan worked with what he called \generalized spaces" (espace g en eralis e). Classically a generalized space was \a space of tangent spaces (which may or may not be the spaces of tangent vectors) such that two innitely near tangent spaces are related by an innitesimal transformation of a given Lie group" i.e., a space with a (possibly nonlinear) connection.[4] At about this same time Weyl had begun his own research into the representation theory of Lie groups. Weyl used the Absolute Dierential Calculus of Ricci, whereas Cartan used his own calculus of dierential forms.[12] The successful generalization of Klein's Erlangen program to innitesimal geometry provided an impressive conceptual frame for studying dierent types of geometries such as Riemannian, Lorentzian, Weylian, ane, conformal, projective, and others. Thanks to Cartan, all of these geometries were not only characterized by connections and curvature but also allowed for the new phenomenon of torsion. 11 Despite the progress being made in the theory of connections, there was still a lack of unication. The ambiguous denition of Cartan's \generalized spaces" did not help to clarify the situation. With the introduction of ber bundles we are nally able to express the concepts related to \connection" in a more satisfactory way and obtain the most general denition. 3.2 Fiber Bundles It was Cartan's student Charles Ehresmann who nally untangled and successfully classied all of the generalized and specic connections which had emerged in the rst half of the 20th century. He began this endeavor in an eort to understand Cartan's connections from a global point of view (as well as being in uenced by Lie and Ernest Vessiot). Toward this end Ehresmann introduced the concept of ber bundles (independently of Whitney and Steenrod). Ehresmann published his rst notes on the subject during the period 1941-1944 in which he denes locally trivial principal bundles and their associated ber bundles (also locally trivial1). In his 1943 paper Sur les espaces br es associ es  a une vari et e di erentiable a manifold is dened by means of an atlas of local charts for the rst time.[8] A main characteristic of principal bundles is that they are ber bundles along with a Lie group Gwhich acts freely on the bers. This group essentially tells what kind of transformations are allowed between the model space Fand the bers. The group Gis referred to as the structure group of the bundle. Thus, a principal bundle is often referred to as a principalG-bundle . An important problem which Ehresmann worked on was reduction of the structure group Gof a principalG-bundle to a subgroup HofG. Ehresmann proved that the reduction of a structure group Gis equivalent to the existence of a section in an associated bundle with standard ber G=H .[8] Furthermore, given a principalG-bundle (P;M;G ) and a subgroup HofG, it can happen that ( P;M;G )=(P;M;H )2. This reduction 1Topological ber bundles over general spaces may or may not be locally trivial. However, ( C1) ber bundles are assumed to be locally trivial since the base space is assumed to be paracompact and paracom- pactness implies local triviality. 2To obtain this equivalence it is necessary that the G-valued cocycle on Mtakes values only in H. 12 of the structure group is such an important concept because the smaller the subgroup, the more restricted are the kinds of transformations which are allowed between the model space and the bers (i.e., the smaller the group structure, the more you know!). 3.3 A Minimal Denition InLes connexions innit esimales dans un espace br e di erentiable , published in 1951, Charles Ehresmann denes an innitesimal connection (known today as an Ehresmann connection ) on a locally trivial ber bundle to be a distribution of n-contact elements (where nis the dimension of the base space) which is transverse to the bers and satises the path lifting property. In the case of a principal G-bundle, Ehresmann adds the condition that \the distribution must be invariant under the right action of G" and goes on to show that \the connection associates an innitesimal displacement to any vector tangent to the base.[8] Ehresmann's concept can be described in modern terms as follows: Let Ebe a dieren- tiable (C1) ber bundle ( E;;M;F ). An Ehresmann connection on Eis a subbundle Hof TE, which is complementary to ker(). In other words, for each v2E,TvE=ker()Hv. The bundle His called the horizontal bundle and the bundle ker() is called the verti- cal bundle. The path lifting property says that given any smooth path onMjoining x0= (t0) tox1= (t1), and anyv02E (t0), there exists a horizontal smooth path joining v021(x0) tov121(x1) which projects on by(i.e.,(v(t)) =x(t) for allt). To see a little more of the situation, consider an open covering UofM. For any open setUinU, there exists a dieomorphism (given by a bundle chart) between 1Uand UF. Thus, each element in 1Ucan be identied with a pair ( p;v);ptells where you are (horizontally) relative to the base Mandvtells where you are (vertically) in the ber over p. Letxibe local coordinates in Uand letyjbe coordinates along the bers in 1U. Then, forp2MjUwe havep= (x1;x2;:::;xn) and for v2Ej1Uwe have v= (y1;y2;:::;yn). Now, for some other open set U02Uwe have (via another bundle chart) coordinates xi0inU0 andyj0along the bers in 1U0. So,p2MjU0impliesp= (x10;x20;:::;xn0) and v2Ej1U0 13 implies v= (y10;y20;:::;yn0). IfU\U06=;say,U\U0=T, then on1Twe havep=xi0(xi) andv=yj0(xi;yj). The choice of a chart induces vector elds@ @xionMand@ @yjonEwith the transfor- mation laws: @f(x0;y0) @xi0=@f(x;y) @xi@xi @xi0+@f(x;y) @yj@yj @yj0;@f(x0;y0) @yj0=@f(x;y) @yj@yj @yj0 Therefore, @ @xi0=@xi @xi0@ @xi+@yj @xi0@ @yj;@ @yj0=@yj @yj0@ @yj and the inverse: @ @xi=@xi0 @xi@ @yj0+@yj0 @xi@ @yj0;@ @yj=@yj0 @yj@ @yj0 Vector elds@ @yjspan what is called the vertical subspace in TE. This subspace, consisting of vectors tangent to the bers, is independent of the choice of chart. There are noa priori distinguished \horizontal" subspaces. The subspaces determined by vector elds @ @xichange when the charts change. A preferred distribution of horizontal subspaces is given by an Ehresmann connection . Once such a connection is given, vector elds@ @xitangent to M can be lifted to horizontal vector elds@ @xitangent to E. In light of Ehresmann's denition, a connection can be seen as a purely geometric object. All the concepts attached to that of a connection can be interpreted in this context and we can make a top down classication of all connections via restrictions on the group G. 14 CHAPTER 4 MODERNITY Thanks to Charles Ehresmann there is now a consensus on what a connection really is (a distribution of horizontal subspaces), as well as a consistent terminology. 4.1 Equivalence At this point the reader might wonder (as I did) how in the world this abstract concept of a particular distribution on a ber bundle is related to the concept of covariant dierenti- ation or parallel transport which have also been (and still sometimes are) called connections. The simple answer is that a system of parallel transport and a covariant derivative operator each result in a distribution of horizontal subspaces. In fact, on any dierentiable ber bun- dle there is a bijective correspondence between a connection Hand a parallelism structure P. On any vector bundle there is a bijective correspondence between a covariant derivative operatorr1and a connection H. For proofs that all three are equivalent geometric concepts see [10]. 4.2 Terminology A connection is said to live \on" a ber bundle or \in" the tangent bundle to a ber bundle. So, a connection on Eis a connection in TE. However, connections were originally studied only in the tangent bundle case and were referred to as on a manifold. This terminology stuck even after the generalization of connections on manifolds to connections on ber bundles, which is why today when the literature refers to a connection on M, one assumes this to mean that the connection is in TTM rather than a more general TEwhich makes sense anyway because if a bundle is not specied, the only one that is naturally present overMisTM. 1In 1950 Jean-Louis Koszul unied the various notions of covariant dierentiation in general vector bundles which arose in the 1940's and also showed a bijective correspondence between randP 15 4.3 Conclusions Before I studied the history of connections I did not understand (nor quite believe) why Dierential Geometry is sometimes described as being \the study of a connection on a principal bundle". The trueness of the statement is now quite obvious. After all, Dierential Geometry takes place on a dierential manifold (which is naturally part of a principal bundle, TM) and a connection is the indispensable structure which allows for both the \dierential" aspect and the \geometry" aspect. 16 REFERENCES 17 LIST OF REFERENCES [1] M.A. Akivis and B.A. Rosenfeld, \ Elie Cartan (1869-1951)", Translations of Mathemat- ical Monographs . vol.123, AMS, 1993. [2] U. Bottazzini, \Ricci and Levi-Civita: from dierential invariants to general relativity", The Symbolic Universe . ed. Gray. Oxford University Press, 241-259, 1999. [3] E. Cartan, \Sur les vari et es  a connexion ane et la th eorie de la relativit e g en eralis ee", Ann. Ecole Norm. Sup. ,40, 1923. [4] S.S. Chern and C. Chevalley, \Elie Cartan and his mathematical work", Bull. AMS ,58, 217-250, 1952. [5] C. Ehresmann, \Les connexions innit esimales dans un espace br e di erentiable", Col- loque de Topologie, Bruxelles , 29-55, Thone, Li ege, 1950. [6] J. Gray, A Century of Geometry . Lecture Notes in Physics, 402, New York: Springer- Verlag, 235-239, 1992. [7] T. Levi-Civita, \Nozione di parallelismo in una variet a qualunque et conseguente speci- cazione geometrica della curvatura riemanniana", Rend. Circ. Mat. Palermo ,42, 173- 204, 1917. [8] P. Liberman, \Charles Ehresmann's concepts in dierential geometry", Geometry and Topology of Manifolds: The Mathematical Legacy of Charles Ehresmann , Banach Center Publications, 76, 2007. [9] L. Del Riego and P.E. Parker, \General connections, exponential maps, and second order dierential equations", Dier. Geom. Dyn. Syst. 13, 72-90, 2011. [10] W.A. Poor, Dierential Geometric Structures . New York: McGraw-Hill, 1981. (Dover reprint, 2007.) [11] K. Reich, \Levi-Civitasche Parallelverschiebung, aner Zusammenhang, Ubertragungsprinzip: 1916/17-1922/23", Arch. Hist. Exact Sci .44, 77-105, 1992. [12] E. Scholz, Weyl and the Theory of Connections, in The Symbolic Universe . ed. Gray. Oxford University Press, 261-284, 1999. [13] M. Spivak, A Comprehensive Introduction to Dierential Geometry , vol.2, Boston: Pub- lish or Perish, 1970-1975. [14] H. Weyl, \Reine Innitesimalgeometrie", Math. Z. ,2, 384-411, 1918. 18 APPENDICES 19 APPENDIX A Note on ane connections When Weyl generalized the Levi-Civita connection to obtain linear parallel transport (with zero torsion) in a vector bundle, he called it an \ane" connection. Shortly afterward, Cartan dropped the symmetry requirement on the Christoel symbols to obtain arbitrary linear parallel transport (with the possibility of torsion) in a vector bundle which he also referred to as an \ane" connection, with the comment that \The expression `ane connec- tion' is borrowed from H. Weyl, although it will be used here in a more general context." Cartan's generalization re ects the role played by the group of ane transformations which can be seen as an extension of the linear group to include translations. In essence, admitting parallel translation as a legitimate motion changes the automorphism group of RnfromGLntoAn. This led to the modern denition of ane connection as a non-linear connection with structure group GLnGAn. Today the term \ane" is used mostly in the sense of Weyl to refer to linear parallel transport (without torsion). However, close attention should be paid to the context whenever the term ane is used since aside from the three formulations given above, there are at least two more (lesser used) denitions of ane connection (see [10]). 20 APPENDIX B Note on Cartan connections In this thesis, I have not delved too deeply into Cartan connections . Cartan con- nections are a specialized type of principal connections which are quite rigidly tied to the underlying geometry of the base manifold by what by what Cartan refers to as a \soldering" of the total space to the base space. All of the technical ways in which Cartan connections are related to other types of connections can be found by reading Cartan's work. But, be fore- warned, as Weyl himself, in reviewing one of Cartan's books, stated \Cartan is undoubtably the greatest living master in Dierential Geometry...I must admit that I found the book, like most of Cartan's papers, hard reading..."[3] a sentiment that I found over and over again in reference to Cartan's work. Thus, I would recommend the following sources for information onCartan connections . S. Kobayashi and K. Nomizu, Foundations of Dierential Geometry . vol.1, New York: Interscience, 1963. A. Magnon and A. Ashtekar, Translation from French of Cartan's work \Sur les Varietes a Connexion Ane et la Relativite Generale" with a Commentary and Foreward by A. Trautman. Bibliopolis, Naples, 1986. R.W. Sharpe, Dierential Geometry . GTM 166. New York: Springer-Verlag, 1997. Corrected second printing, 2000. 21 APPENDIX C Chart of connections (general) connection H ?+ HPL Ehresmann connection ?+ a Lie group Gwhich acts on F i.e., the splitting is equivariant G-connection ?+ the action of GonFisfree principal connection (i.e. principalG-connection) ? ? linear connection+GGLn non-linear connection+GGLn ? ?+GAn ane connection (Cartan/Ehresmann)+GOp q pseudo-Riemannian ?+ 0torsion connection ? Riemannian connection metric connection+GOn + 0torsion ? Levi-Civita connection+F=Rn \ane" connection (Weyl)+ 0torsion ? 22