CONNECTIONS ON PRINCIPAL BUNDLES
by Matt Lewis
SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE
AT
DALHOUSIE UNIVERSITY HALIFAX, NOVA SCOTIA DECEMBER 2007
(c) Copyright by Matt Lewis, 2007 Library and Bibliotheque et 1*1 Archives Canada Archives Canada Published Heritage Direction du Branch Patrimoine de I'edition
395 Wellington Street 395, rue Wellington Ottawa ON K1A0N4 Ottawa ON K1A0N4 Canada Canada
Your file Votre reference ISBN: 978-0-494-39133-4 Our file Notre reference ISBN: 978-0-494-39133-4
NOTICE: AVIS: The author has granted a non L'auteur a accorde une licence non exclusive exclusive license allowing Library permettant a la Bibliotheque et Archives and Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par telecommunication ou par Plntemet, prefer, telecommunication or on the Internet, distribuer et vendre des theses partout dans loan, distribute and sell theses le monde, a des fins commerciales ou autres, worldwide, for commercial or non sur support microforme, papier, electronique commercial purposes, in microform, et/ou autres formats. paper, electronic and/or any other formats.
The author retains copyright L'auteur conserve la propriete du droit d'auteur ownership and moral rights in et des droits moraux qui protege cette these. this thesis. Neither the thesis Ni la these ni des extraits substantiels de nor substantial extracts from it celle-ci ne doivent etre imprimes ou autrement may be printed or otherwise reproduits sans son autorisation. reproduced without the author's permission.
In compliance with the Canadian Conformement a la loi canadienne Privacy Act some supporting sur la protection de la vie privee, forms may have been removed quelques formulaires secondaires from this thesis. ont ete enleves de cette these.
While these forms may be included Bien que ces formulaires in the document page count, aient inclus dans la pagination, their removal does not represent il n'y aura aucun contenu manquant. any loss of content from the thesis. Canada DALHOUSIE UNIVERSITY
To comply with the Canadian Privacy Act the National Library of Canada has requested that the following pages be removed from this copy of the thesis:
Preliminary Pages Examiners Signature Page (pii) Dalhousie Library Copyright Agreement (piii)
Appendices Copyright Releases (if applicable) Table of Contents
Abstract vi
List of Abbreviations and Symbols Used vii
Acknowledgements ix
Chapter 1 Introduction 1
Chapter 2 Maurer-Cartan Form and Principal Bundles 4
2.1 The Maurer-Cartan Form 4 2.1.1 The Adjoint Action 4 2.1.2 The Frolicher-Nijenhuis Bracket 5 2.1.3 The Maurer-Cartan Form 7 2.1.4 The Darboux Derivative 9 2.2 Fibre Bundles 10 2.2.1 Bundles 10 2.2.2 Principal Bundles 13 2.2.3 Associated Bundles 15 2.2.4 The Frame Bundle . 18
Chapter 3 Connections 20 3.1 Connections 20 3.1.1 Ehresmann Connections 20 3.1.2 Cartan Connections 26 3.1.3 Existence of Cartan Connections 29 3.2 Cartan Geometry 36
Chapter 4 Ehresmann Connections versus Cartan Connections 39 4.1 Parallel Displacement 39
IV 4.1.1 Ehresmann Parallel Displacement 39 4.1.2 Cartan Parallel Displacement 40 4.2 Holonomy 44 4.2.1 Ehresmann Holonomy 44 4.2.2 Cartan Holonomy . 45 4.2.3 Ehresmann Holonomy as Cartan Holonomy 49 4.3 The Covariant Exterior Derivative . 51
Chapter 5 Conclusion 59
Bibliography 62
v Abstract
This thesis is devoted to the study of connections in geometry, in particular, on principal bundles. The two types of connections on principal bundles, Ehresmann, and (generalized) Cartan connections, are introduced and the theory of the corresponding geometries are developed. Finally the two connections are compared, for one to see the advantages of having a Cartan connection over a Ehresmann connection.
VI List of Abbreviations and Symbols Used
Sn the symmetric group on n elements
Euc„ Euclidean group acting on Rn
O(n) orthogonal group acting on W1
SO(n) special orthogonal group acting on W1
G K H semi-direct product of two groups G and H
M,N Smooth, paracompact manifolds
G,H,L Lie groups
g, f), p Lie algebras to the Lie groups G, H, and L respectively
Rh, Lh left and right translations by an element h of a Lie group
C(g) the adjoint action of an element g in a group Lie group G
cl(G) the closure of a topological group G
Ar(M) the set of differential r-forms on M
Ar(M, Q) the set of g-valued r-forms on M
X^ fundamental vector field corresponding to a vector X € 0
[-,-] ...... Frolicher-Nijenhuis bracket
[-, -]8 the commutator on the Lie algebra g
U>G Maurer-Cartan form on a Lie group G
/* push-forward induced by a diffeomorphism /
/* pull-back induced by a diffeomorphism /
vii u)f Darboux derivative with respect to a map / : M *• G
G >• P v > M right principal G-bundle
P[F] associated bundle to P with fibre F
7 Ehresmann connection ijj Cartan connection
W generalized Cartan connection
$![•] curvature of the corresponding connection
(P, 7) Ehresmann geometry
(P,u>) Cartan geometry
(P,oJ) generalized Cartan geometry r(/\r T*P, V)G set of V-valued pseudotensorial r-forms on P r\or(Ar T*P, V)G the set of I^-valued tensorial r-forms on P
9 soldering form c horizontal lift of a path c c development of a path c
dy covariant exterior derivative with respect to 7 d^ covariant exterior derivative with respect to OJ
dw covariant exterior derivative with respect to a;
vm Acknowledgements
I would like to thank my supervisor Roman Smirnov for all his help and encourage ment. I would also like to thank my other two readers Ray Mclenaghan and Rob Milson for agreeing to read this thesis.
ix Chapter 1
Introduction
The history of geometry has gone through three major generalizations from Euclidean geometry. The first generalization came in a series of lectures given by Riemann, at the University of Gottingen in 1854. This "new" manner in regarding geometry, was inevitably entitled Riemannian geometry, and relies entirely on the concept of differentiable manifolds, which is simply a generalization of Euclidean space. The key concept being generalized, is the notion of curvature, where we leave the realm of geometric objects having zero curvature, and proceed in the direction of non-zero curvature, both constant and non-constant.
After Riemann, came Klein, who attempted to answer the question "what is ge ometry?" , by claiming that "geometry is the study of geometric objects and their in variant properties under a given transformation group". He explored this idea in 1872 in what is known as "Klein's Erlangen Program". This development uses the ideas of Riemann, namely differentiable manifolds, and the concept of Lie groups. In the case n of Euclidean geometry, we consider the Euclidean group Eucn = O(n) ix M (or in the case of preserving orientation, the special Euclidean group SE(n) = SO(n) K R"), from which we can form Euclidean space by taking the quotient space Euc„/0(n) (SE(n)/SO(n)). Thus the corresponding symmetry group is given by O(n) (SO(n)). The manner in which Klein effects this generalization, is by considering an arbitrary Lie group G, with a closed subgroup H C G, and look at the space G/H, from which we see the symmetry group will be given by H, that is, we simply generalize the symmetry group from O(n) (SO(n)).
The notion of Klein geometry gives rise to the idea of connections on principal bundles, which in this case is simply the Maurer-Cartan form, u>a, on the group G, which satisfies the equation du>o + OJQ AWG = 0 (the structure equation). The final major generalization came from the ideas of Cartan in the early 1920's, who
1 2 joined Klein's ideas and Riemann's. That is, we generalize the symmetry group, and add curvature. The curvature in the sense we speak of here, is the curvature of the the connection on a principal bundle. To study the ideas of Cartan, we need the concept of a Cartan connection, say to, on a principal bundle H »- P v > M. If we compute the value du> + cu AOJ, it will in general be non-zero, in contrast to the value zero for the Maurer-Cartan form. The "curvature" of the connection is given by Q[LO] = dio + u> A u, and can be said to measure the "lumpiness" of the space in question, or simply the failure of the structure equation. The above describes the generalization from Klein geometry to Cartan geometry. The manner in which Cartan geometry generalizes Riemannian geometry, is that the tangent space is generalized. With aid of a Cartan connection on a principal bundle H >• P n > M, we may take the Klein geometry G/H, and "roll" it on the underlying geometry M in question. This connection allows this Klein geometry to roll with "slipping" or "twisting", for an accurate description of M. These three major generalizations can be summarize by the following diagram, adapted from [8] and [9],
Euclidean Generalize Klein Geonaetr y Symmetry Group Geonnetr y
Add Add Curvature Curvature
• • Riema tan tnnian Generalize Car
—• =-- Geometry Tangent Space Geometry
In the process of defining the Cartan connection, which is what Cartan ultimately wanted, there was the Ehresmann connection, developed by Cartan's student Charles Ehresmann. At first glance, these connections seem unrelated, however if we pass from the original principal bundle, to an associated bundle, we achieve some in teresting results. The main result is that there is a restriction to the Ehresmann connections on the associated bundle, such that these are in bijective correspondence 3 with the Cartan connections on the original bundle in question. This thesis develops the theory of the two types of connections, explains the advantage of having a Cartan connection over an Ehresmann connection, and shows how they correspond with par allel transport, holonomy, and covariant exterior differentiation under this particular correspondence. Most of the notation used is compatible with that of [8] and [4]. Furthermore, all manifolds, in particular, the base spaces of principal bundles are assumed to be paracompact, as this importance becomes clear with Theorem 3.1. Chapter 2
Maurer-Cartan Form and Principal Bundles
All the material in this thesis relies heavily on the understanding of principal bun dles. As a consequence, we must understand the adjoint action (representation), Frolicher-Nijenhuis bracket, Darboux derivative, and the theory of general (smooth) fibre bundles. This chapter is devoted to the understanding of such material, and finishes with a specific type of fibre bundle which can be constructed from an original one, this new bundle being called the associated bundle. The first section is devoted to the Maurer-Cartan form, and the second is to fibre bundles.
2.1 The Maurer-Cartan Form
2.1.1 The Adjoint Action
The adjoint action serves the same purpose as the action defined by conjugation, to detect non-commutativity of a group. Let G be a Lie group, and let C : G x G >• G denote the action defined by conjugation, i.e.
C.GxG- — G
(g,h)\ *C(g)h = ghg-1.
This map induces the homomorphism of the Lie group for each g € G,
C(g):G -G
h i •• ghg~l
4 5 called the inner automorphism induced by g. The adjoint action is defined by,
C:G ^Aut(G)
9i ^C(g).
The derivative of C at e E G (the identity), is of utmost importance in the theory of Lie groups and Lie algebras.
Definition 2.1. Let G be a Lie group, with Lie algebra g. Define Ad(g) = C(g)*e € Gl(g). The (left)-adjoint representation is defined as the map
Ad : G > Gl(g)
9 ^Ad(g)
Equivalently we may speak of the (left) adjoint action ofG ong as the map Gxg >- g sending (g,X) t-> Ad(g)X.
Remark 2.1. For a vector X € g, the adjoint action on X is then given by Ad(g)X =
2.1.2 The Frolicher-Nijenhuis Bracket
The Frolicher-Nijenhuis bracket, is an operation on vector valued differential forms. T s Let M be a manifold, g a Lie algebra, and UJ\ G A (M,g), UJ2 € A (M,g). The Frolicher-Nijenhuis'bracket is a mapping [•,•] : Ar(M,g) x As(M,g) »- Ar+s{M,g) defined by 6
[uj1,u2](Xi,...,Xr+s)
= -j-j- 2_/ ^g^M^iP^Ki)' • • • >X^r)),U2(Xa(r+i),.. .,Xa(r+s))] / .6. Si n a; + r|/ _ ^| 2 § (°") 2([^l(^cr(l), • • • ,Xo{r)),Xu{r+1)},Xa{r+2, ...) '*• '" cr&Sr+s + (r-l)!s! Yl nm(<7)ui([u2(Xa{1),_...,X„(a)),Xe{a+i)],Xa{a+2),...)
cr€Sr+s + (r-l)!(s-l)!2! 5Z &&(a)^M[x
+ (r-l)!(s-l)!2! ^ ^m{
r S Proposition 2.1. Let ui E A (M) and u2 € A (M). Then the Frolicher-Nijenhuis bracket satisfies the following properties.
+1 1. [a;1,W2] = (-l)" N,wi]
2. Ifr = s = l, then
[uuu2](X,Y) = [Wl(A-),^(y)]B + [w2(X),c«;i(y)]0 = Wl A^(X,r)
Proof. These follow directly from the definition of the Frolicher-Nijenhuis bracket. •
From property (1), of proposition 2.1, we have an analogue of the Jacobi identity: p q r Let up,Ug,ujr E A ' ' (M,g), then
r p
Corollary 2.1. If ui € A(M,g) then
[u>M(X,Y) = ±[u(X)MY)]9.
Proof. Follows from property (2) of the Frolicher-Nijenhuis bracket. •
These concepts will become of importance in the study of connections, in par ticular, the curvature of a connection. The brackets will be identified as [•, •] as the
Frolicher-Nijenhuis bracket, and [-,-]g as the Lie bracket on a Lie algebra 0 unless otherwise stated. 7
2.1.3 The Maurer-Cartan Form
For every Lie group G, there exists a g-valued one-form u>G : TG *• Q satisfying the Maurer-Cartan equation
du)G + -[wG,wc| = 0, called the Maurer-Cartan form. The Maurer-Cartan form takes vector fields on G, and transports them to Q — TeG. With this in mind, we define the (left-invariant) Maurer-Cartan form as
UG{X) = Lg-l*X\g, from which we see LOQ gives an isomorphism from the space of all left-invariant vector fields (i.e. the vector fields X\g £ TgG satisfying Lh*X\g = X\hg) to g.
Remark 2.2. The term left invariant when defining the Maurer-Cartan form OJG on a Lie group G, comes from the fact that if, X G TgG, then for any h G G
L*huG(X) = ujG(Lh*X) = L(hg)-1*Lh*X = Lg-i^Lh-i^Lh^X
= Lg-lifX
= UG(X).
Definition 2.2. Let G be a Lie group, and M a smooth manifold. If G acts on M smoothly, then M is said to be a G-space.
Proposition 2.2. Let G be a Lie group, M a G-space, /J, : G x G *- G denote group multiplication, and t : G »- G denote group inversion. Then u>c satisfies the following properties:
l 1. R*hWG — Ad(h~ )ujG for all h G G, i.e. UQ is equivariant
x 2. n*u}G = { 3. i*uG — —Aa\oG. 4- For fa, fa :M *~ G, define h(x) = fa(x) fa(x). Then 1 h*LuG = Ad{fc )flujG + fZujG. Proof. (1) Let X e TgG, then R*huG(X) = ojG{Rh*X) — L(ghyi*Rfi*X = L^-i^Lg-i^R^X = Adih-^uoiX), x Therefore R*hooG — Ad{h~ )uG. The details for the proof (2) and (3) are technical, and of not that much importance, the reader is referred to [8]. For (4), write h as the composition M—^MxM hxh > GxG »G xi ^ (x, x) i ^ {fi(x), fa(x)) ^—+ fa(x)fa(x) Then with h = JJ, O (fx X /2) o A, we have h*ujG = (/io (fi x /2) o A)*wG = ((/i x/2) o A)VW = ((fa x fa) ° A)* [TrKAd)-1^*^ + TT^UCI] (from property 2) 1 = fa o (fa x /2) o AY(Ad)- (7r1 oXfa x fa) o A)*UJG +fa O (/i x /2) o A)*u;G i = (/2)*(Ad)- /r^G +/>G = Adif^fivG + fiUG. 1 Therefore h*ujG = (fafa)*uG = Ad^" )/;^ + />G. Another property that will become useful in the following chapters is that, if H and G are Lie groups, with cl(iJ) = H and H C G, then u/# = WGIH• We can in fact generalize this idea as follows. Proposition 2.3. Let cp : G\ *• G2 be a homomorphism of Lie groups, ui\ = ucl} l UJ and UJ2 = Maurer-Carian form on G2 to G\ via ip, and the right hand side is the Maurer-Cartan form on G\ with values in Q2 via Proof. Since cp is a homomorphism of Lie groups, for every g € G\ the following diagram commutes: v(sr) Thus if X E T„Gi, we have ( = = <£>*e(cJi(X)). D To see that this proves our claim of WH = U>G\H, we have that H is a Lie subgroup, since it is closed, and we can take (p : H «—»• G to be the inclusion map. The result now follows. 2.1.4 The Darboux Derivative The Darboux derivative generalizes the notion of the usual derivative of functions on 3EL The particular interest of the Darboux derivative, is that it provides us with a sufficient number of invariants of a manifold under a particular group action. Definition 2.3. Let M be a manifold, G a Lie group with Lie algebra g, and f : M >- G a smooth map. The Darboux derivative of f, is the g-valued one-form uif : TM 9- Q defined by uif = f*u>a- 10 A more visual way to look at the Darboux derivative, is to consider the composi tion: cuGf* = f*uG = ujf:TM *TG -0. From the commutativity of the following diagram Ar(M) ^— Ar+1(M) r Ar(G) where ip : M -*- G is a smooth map, and the structure equation, we have the relation dcjf + -[wf,uf]=0. Furthermore, if f\, ji : M *• G are two smooth maps with tu/j = u>/2, then fzix) — c' f\{x)i where c € G is some constant, which can be thought of as the "constant of integration". Therefore the change in Darboux derivative, that is, the change from u)fx to djf2 can be related by the familiar equation 1 u)h = Ad(c~ )ujh + fiwfl. Thus the Darboux derivative is unique for each M, up to this transformation. 2.2 Fibre Bundles 2.2.1 Bundles A general bundle is triple (E, n, B) consisting of two topological spaces E and B, called the total space and base space respectively, and a projection IT : E >• B. An additional structure is found from this called the fibre Ff, := 7r_1(6), for b 6 B. The best intuitive manner in looking at a fibre bundle, comes from [6]: "intuitively one can think of a bundle as the union of fibres, 7r_1(6) for b € B, parametrized by B and "glued together" by the topology of E". Of particular importance are fibre bundles, which are bundles but with added structure. A fibre bundle is, so to speak, 11 a topological space which locally looks like a direct product of two topological spaces. Now to a more formal definition. Definition 2.4. Let F be a topological space and n : E >• B a continuous surjective map, called the projection. The quadruple (E,B,TT,F) is called a fibre bundle with fibre F if, for each point b € B, there exists an open subset U C B such that b € U and TV~1(U) is homeomorphic to U x F via a homeomorphism ip such that the following diagram commutes. n-\U) -UxF where TV\ denotes the projection from the first coordinate. The fibre at the point b e B is given by Fb = 7r-1(&) and is homeomorphic to F, which can be regarded as F = U6eS JFJ,, the disjoint union of the fibres over B. E and B are called the total space and base space respectively. In the case where E, B and F are smooth manifolds, ir is smooth, and the ip as above is a diffeomorphism, then we have a smooth bundle. A (smooth) bundle is said to be trivial if E is (diffeomorphic) homeomorphic to F x B, E = F x B. For this reason, we call the function tp in the above a local trivialization. The notation E n > B may also be used to denote a bundle. Definition 2.5. Let E —*• B be a smooth bundle with fibre F, and suppose that G C Diff{F) is a Lie group. A G atlas for the bundle is a collection {(Ui, (p)} of charts such that 1. the f/j's cover B 2. for each pair of charts (U, ip) and (V,ip) in the atlas, the map $ = -0 o tp-1 : (U D V) x F —+ (U n V) x F, called a coordinate change, has the form $(«,/) = (u,h(u)f), where h : U D V ——*• G is a smooth map called a transition function. The group G is called the structure group of the bundle. 12 To achieve a more visual understanding of the connection between fibre bundles and G atlases, we have the following commutative diagram: (unv)xF^—Tr-1{unV)-^(UnV)xF unv where ip and ip are the homeomorphisms: ir~l(U) ~ > U xF and 7r_1(V) ~-> VxF respectively. Note the abuse of notation in the above diagram, the mappings (p and ip are actually the mappings restricted to the non-empty intersection U D V. That is, in the diagram, Definition 2.6. Let E —^-»- M be a smooth bundle. A smooth map s : U - *• E is called a section if IT O S = id,M, for some open subset U C M. Example 2.1. The simplest example of a bundle, is the tangent bundle to a smooth manifold M. Let X G TXM. Define E := TM := UxgM TXM, and TT : E ^ M as 1 l X i—» x. Then 7r~ (x) = TXM = Fx is the fibre. Using local coordinates (x ) on M, we may write X as _d_ X = X{ dxi If (yl) are some other coordinates on M, transformed from (xl) by some diffeomor- phism f, then we have j X*± = # £ X = X i dxl dyJ dx Since f is a diffeomorphism, the Jacobian (dyi/dx1) is non-singular at each point, and hence lives in Gln(M.), where we see the G-atlas consists of local coordinate charts, and the general linear group Gln(M). In this setting, vector fields may be viewed as sections of the tangent bundle. 13 2 Definition 2.7. Let Ei,E2 •— > M be two fibre bundles with same base space M. A bundle isomorphism, is a diffeomorphism ip : E\ —r-s- E2 such that the following diagram commutes A vector bundle is simply a fibre bundle with the fibre being a vector space. As we have seen above, the tangent bundle forms a vector bundle, since TXM is a vector space for all x G M. Similar to the tangent bundle we have the cotangent bundle, T*M, which is constructed in an analogous way. Here, the sections are given by the one-forms on M. In analogy with bundle isomorphisms, a vector bundle isomorphism, is a bundle isomorphism between vector bundles, which is also linear on each fibre. 2.2.2 Principal Bundles As one would expect from the introductory chapter, and the claim that "Differential geometry is the study of connections on principal bundles", principal bundles carry a wealth of information. A principal bundle is, loosely speaking, a smooth bundle where the fibre can be identified as the structure group. Definition 2.8. A principal G-bundle is a smooth bundle (P, M, n, F) together with right action P xG *• P that is fibre preserving and acts simply transitively on each fibre. Remark 2.3. A simply transitive action is an action that is free and transitive. To denote a principal G-bundle, we will write G >- P * > M. The superscript iv in G >• P —^-*- M will always be included, to avoid confusion with short exact sequences. However, the principal bundle notation can be taken to be a short exact sequence by letting % : Gc *• P be the inclusion map, then the short exact sequence is given by 0 ^ G —U P -JU- M = P/G *• 0, 14 since G = ker(7r). Given such a bundle, it can be shown that the action Gx P »- P is proper, i.e. given two compact subsets A,B C P the set {g & G : (gA) n B / 0} is compact. Example 2.2. Consider the case where we have a Lie group G and subgroup H C G, such that cl(H) = H. To define a principal H-bundle from this data, we take G and project it on the orbit space G/H, that is define the map IT : G >• G/H by sending g i—> gH, which is simply the natural projection. The fibre of this map is H itself. The transition functions are given by elements of H, and since H is the isotropy group of G/H, we have the principal bundle H »- G —^-»- G/H. For a more visual understanding of the example above, one can take G — SO(3), and H — SO(2). Under these circumstances, the orbit space G/H can be identified with §2, from which we see the isotropy group of this is H = SO(2). The example above is very important, as it gives rise to the notion of a "Klein geometry", which is a very fundamental concept when wanting to study Cartan connections. Another application of principal bundles, has to do with a classification problem, namely, determining whether or not a bundle is trivial. Proposition 2.4. Let G *• P * > M be a principal bundle. A global section s : M *- P exists if and only if P is trivial. Proof. (=»): Let s : M -—*- P be a global section, and g G G. Then the element s(x) • g lies in the fibre at x, and since G acts on P simply transitively, and element of p G P can be uniquely written as s(x) • g, for some x G M, g G G. Now define the map $ : P *- MxGby s(x) • g »->• (x, g). This is easily seen to be a homeomorphism. Therefore P = M x G. (<£=): Suppose P is homeomorphic to M x G. Let cp be a trivialization of P. Fix g G G, and define the map sg : M >• P by x i—• We finish this section with the following definition. 15 Definition 2.9. Let H be a Lie subgroup of G. Then G *- P n > M, is called H reducible, if there exists a principal bundle H >• Q ——>- M and an H equivariant embedding i: Q *- P, i.e. an embedding satisfying t{qh) = t(q)h for all q € Q and h G H. Then Q is called a H-reduction of P. 2.2.3 Associated Bundles The study of principal bundles with fibre and structure group G, allows us to construct a bundle with same base space and structure group, but different fibre. That is, given a G-bundle G -—>• P —?-*- M, we are able to construct a second bundle E »- M with a different fibre, F, but with the same structure group G. This new bundle is called an associated bundle to P. The construction is as follows: Let G *- P —^-*- M be a principal G-bundle, and F a manifold on which G acts on the left. Consider the product space Px F and x define the action of g € G on (p, /) e P x F by (p, /) t-> (pg: g~ f). The associated bundle is then an equivalence class (P x F)/G in which points (p, f) and (p^,^-1/) are identified. Note also that if we are given a vector bundle, we may construct a principal bundle, which is associated with the original bundle. Let E >• M be a vector bundle with dim(£;) = k, then E induces a principal bundle over M having the same transition functions, and the structure group is given by GL(k) over R or C depending on whether the fibre is Kfc or Cfc. For p e P and f £ F denote the orbit space of (p, f) by b./]:=P/:={(P<7. We may then define the associated bundle to P n > M by PxGF:=(PxF)/G = {pf:peP, f e F} . Note that for every p & P with 7r(p) = x, p defines a diffeomorphism F *FX:=TT-\X)- p: < f •-*• Pf- Given a principal G-bundle G *- P n > M, we will denote the associated bundle with fibre F as P[F] := P xG F. 16 Proposition 2.5. Let G >• P n > M be a right principal G-bundle, and F a G-space via the action p : G 5- Aut(F). Let be P[F] the associated bundle with fibre F. Then there is a bijective correspondence between the following: 1. Global sections on P[F] 2. Smooth, G-equivariant maps f : P >• F, that is, maps f : P >• F satisfying f(p-9) = p(g-1)f(p)- Proof. By definition of a principal bundle, we may identify M as the quotient space P/G. Define the cross section s on P[F] corresponding to / as s/(xG) — (x, f(x))G in P[F] for each xG e M. Since (pg,f(pg))G = (pg;p{g-1)j\p))G = (p,f(p)) in P[F], the function Sf is well-defined. To see that Sf is smooth we notice that Sf is the factorization of p 1—• (p, f(p))G, by the projection IT on P, therefore Sf is smooth. If we denote irp '• P[F] >• M as the defining fibration, we have that n F(sf(pG)) = TTF((P, f(p))G) — 7r(p) = pG holds, and hence 5/ is a cross section. Conversely, let s be a global cross section on P[F], and fs : P *- F be de fined by the relation s(pG) = (p, fs(p))G for each p £ P. Since (p, fs{p))G — l (P9ip{9~ )fs{jp))G — (pg, fs(pg))G, and since s is a smooth cross section, fs sat l r isfies the relation fs(pg) = p{g~ )fs{p) f° each p G P and g G G. Since / 1—> Sf and s \—> fs are inverses, we have that the bijective correspondence is satisfied, which finishes the proof. • Of particular interest in this thesis, is the case when we have as the two fibers, two Lie groups H and G, with H C G. That is, given a right principal iJ-bundle H >• P —?-*- M, we wish to construct the associated bundle P[G], with new fibre G, There are some interesting consequences of this, which will be explored later in the thesis. For now, let us construct the associated bundle P[G] from H >• P * > M. Consider the projection q : P x G >• P xH G, where the action of H on G is given by left translation. This map is denned by (p, g) 1—> \p, g] = pg and gives rise to the principal bundle H ^PxG-^PxHG. The map q, along with the original fibration n : P >• M, and ixp : P x G *-P as the natural projection onto P, induces a unique projection projP : P xH G >• / 17 projF P/H ^ M making P xHG P/H a principal G bundle. We summarize as follows G H -PxG PXHG 7Tp projp H- -^P -^ P/# ^ M The bundle G PxHG P/H is simply the associated bundle P[G]. As a consequence of this map q, we get a special condition on the vector fields of P x# G. _1 Since q(r(p, h),g) = g(p, /i #), we have g*(r*(Xp, Zh), Yg) = q*(Xp, Lh*Yg + Rg*Zh), where r : P x H »- F denotes the right action. This gives rise to the right action r:P[G] xG >P[G\, defined by (q(p,g),g') *-> q(p,gg')- Furthermore, since H C G is equivariant, we get a homomorphism of principal bundles over H C G To finish off this section, we have the following definition, which is very important for the next chapter. Definition 2.10. Let M be an n-dimensional manifold. We call the principal bundle H *• P —^-*- M geometrizable if the tangent bundle TM is associated to P. That 1 is, if there exists a representation p : H *- Gln(M) turning W into an H-module, such that the associated bundle satisfies the soldering condition: n PxHR ^TM. 18 If P is faithful (injective) then P is first order geometrizable. Once a representation exists that satisfies the soldering condition, P is said to be geometrical. 2.2 A The Frame Bundle Given any vector bundle, one can construct a principal bundle called the frame bundle. A frame at a point p £ M is an ordered basis for the corresponding fibre Vp. The set of all frames at the point p, denoted by Fp has a natural free, transitive right action by GLk(M). Now define L(M) — \\peM Fp. We then have the principal bundle GLfc(M) *• L(M) —^-*- M, where if X is a frame at p, -K is the natural projection, (X,p) i—> p. A frame is then given by a section of the frame bundle. Definition 2.11. The frame bundle of a smooth manifold M, is the bundle associated with the tangent bundle TM —->- M as constructed above. Similarly to the relation between the tangent bundle and cotangent bundle, given a frame bundle, we may construct a dual bundle called the coframe bundle. The advantages to using frame and coframe bundles, is that it allows us to work with arbitrary frames rather than coordinate frames. Thus one may adapt such a frame to the geometrical object of interest, and later introduce coordinates. Example 2.3. Consider the Riemannian manifold (En,8ij), and the Euclidean group EuCn = 0{n) tx Rn. The group EuCn is a matrix Lie subgroup of GL(n + 1,M) and can be represented as a group of (n + 1) x (n + 1) matrices: Eucn=<( ) :Ae 0(n),6eEnl. By representing a vector x G E" as an (n+ 1) dimensional vector I L we define a (left) group action p : Eucn x E" >- E" by d)-(0)-(T)- A frame on any n-dimensional Riemannian manifold M is a (n + l)-tuple (e1,e2,...en,x), 19 where x G M and {ei,e2,..., en} is a basis for TXM. Thus by letting (e1;e2,... ,en,x) n be an orthonormal frame on E ; and by defining thenxn matrix A = [ei, e2,.. -, en], we have that A G 0(n) (since {ei,e2,... ,en} is an orthogonal set). In this manner we consider the vector x € En as the translational component of the frame. In light of the above discussion it is seen that there is an isomorphism between the n n frames on E™ and Eucn. Let F denote the collection of frames on E , and define the n map 7r: F >• E" by {e\, e2,..., en, x) H-> X. The fibre of this map is given by _1 n 7r (a;) = {(ei, e2,..., e„) : {e~\, e2,..., en} is an orthonormal basis for TxE } . _1 By defining a function a : 7r (x) »- 0(n) by cr((ei,..., en)) = [ei,..., en], we have that with componentwise addition on the left hand side, a defines an isomorphism, and thus 7T~1(x) = 0{n). In this way we have defined EuCn as a principal fibre bundle with structure group 0(n), called the frame bundle over E" Chapter 3 Connections Note on References: An excellent source for material on Ehresmann connections is found in [7]. For Cartan connections and Cartan geometry, the main sources of reference are from [8] and [9]. In this chapter, we introduce the notion of an Ehresmann connection, and a Cartan connection on a principal bundle. At first the two types of connections seem unre lated; however, the next chapter will explore the similarities. The study of Cartan connections on principal bundles, gives rise to the idea of rolling a "nicer" manifold on another, to study the geometry of the underlying manifold. This "nicer" manifold gives rise to the notion of a Klein geometry, which is essentially just the flat version of a Cartan geometry, as is seen in the development of geometry in the introductory chapter. This idea also generalizes Riemannian geometry in the sense of general izing the tangent space, which becomes this rolling Klein geometry, rather than a generalized tangent plane from 3-space. Before discussing Cartan connections, we must explore a more general type of connection, known as the Ehresmann connec tion. Sometimes this is referred to as simply "a connection on a principal bundle", or a "linear connection". 3.1 Connections 3.1.1 Ehresmann Connections In the study of Differential Geometry, a connection is often defined axiomatically in terms of its covariant derivatives, so in turn we define it as a differential operator which takes directional derivatives of sections of a given vector bundle. Ehresmann connections are defined in terms of the sections parallel in each direction. In a similar manner to Riemannian geometry we are able to give notions of parallelism. Using 20 21 the connection in the Riemannian geometry sense, suppose we have a vector bundle 7T : E »- M, then a section s : M -—3- E is parallel to a vector V E F if Vys = 0. Ehresmann connections, which are defined on any bundle, drop the notion of a differentiable operator, and say that section s is parallel in the direction of V if ds(V) — 0. In this setting we say that s is horizontal in the direction of V, and that ds lies in a horizontal subspace of the tangent space TPM that V lives in. For a more precise definition we have: Definition 3.1. Let G *• P —^->- M be a principle bundle. For each p E P, let Vp denote the subspace of TpP consisting of vectors tangent to the fibre through p, that is Vp = ker(7r*j,). A connection F in P is an assignment of a subspace Hp ofTpP to each p E P such that 1. TP(P) = HP®VP for everyp£P 2. (Rg)*Hp = Hpg for every p E P and g E G 3. Hp depends differentiably on p. The subspaces Hp and Vp are called the horizontal and vertical subspaces of TPP respectively. Condition (1) says that given any vector X E TPP we can uniquely express this as X = hX + vX, where hX E Hp and vX E Vp. The vectors hX and vX are called the horizontal and vertical components respectively of X. Condition (2) implies that the distribution p ——»- Hp is invariant under G. Condition (3) means that if X is a differentiable vector field on P then so are hX and vX. Definition 3.2. Let F be a connection in G >• P >• M, a connection 1-form 7 for T is a 1-form which takes value in the Lie algebra QofG where for each X E TPP', 7p0 = A € Q where (A^)p = vX. Since every A E 0 induces a vector field A^ on F, called the fundamental vector field corresponding to A, we know that the connection 1-form is well defined. By the definition of a connection on a principle bundle, the vector A in the above definition is unique. 22 Proposition 3.1. The connection 1-form 7 of a connection T in P satisfies the following conditions: 1. 7(A) = AforallAeg 2. j(X) = 0 if and only if X is horizontal 1 1 3. (Rg)*j = Ad(g~ )j) or equivalent^ j((Rg)*X) - Ad(g~ )j(X), for every g G G and X G TP, which is equivalent to saying that 7 is G-equivariant. Proof. Let 7 be a connection 1-form of a connection F. Property (1) follows immedi ately from the definition of 7. For (2) suppose X G TPP, then -y(X) = 0 implies that 0 = (0t)u is the unique vertical component of X, hence X is horizontal. Conversely, if X is horizontal then it follows that j(X) = 0 since X does not have a vertical component. To prove property (3), we note that from the definition of 7 we can decompose any vector into its vertical and horizontal components, so it is enough to split this up into two cases. First suppose X is horizontal, then it follows that for any 1 / g G G, (Rg)*X is horizontal thus j((Rg)*X) = Ad(g~ ) y(X) — 0. Now suppose that X is vertical, again using the definition of 7 we may assume that X is a fundamental vector field A^. Claim: X is the fundamental vector field corresponding to Ad(<7-1)A Proof. Let X be the fundamental vector field corresponding to A G Q. Since X is in duced by the group of transformations Rgt where gt — exp(tA), the vector field (Rg)*X is induced by the one parameter group of transformations RgRatRg-i = Rg-igtg- Now we note that g~xgtg is the one parameter subgroup generated by (Ad(<7-1))yl G g. • Hence, from using the claim we have (R*g Conversely, we have that every one form satisfying these conditions gives rise to a connection, to see this we simply let Hp = ker{^p) and we can easily see that this satisfies the properties of a connection. Thus we usually refer to a connection as the connection one form. The connection referred to in the above is known as the Ehresmann connection. The relationship between T and 7 can be represented as follows: r(u,v) • -< v 7(11,11) U The geometric interpretation of this diagram, is that 7 acts inside the fibre (ver tically), whereas T defines a notion of horizontality. We may now reformulate the definition of an Ehresmann connection in terms of the connection 1-form. Definition 3.3. Let G ——*- P —^-»- M be a principal right G-bundle, an Ehresmann connection on P, is a g-valued one form 7 : TP >• g, satisfying the following conditions: 1 1. Rg*-y = Ad(g~ )'j for all g e G. 2. 7(Xt) = X for all X € fl. ' The curvature of the connection 7 is defined as ft [7] =d7 + -[7,7]. The curvature takes values in the Lie algebra g, and given two horizontal vector fields X, Y on P, Q(X, Y) yields the vertical component of the Lie bracket. The geometric interpretation of the curvature is closely related to the notion of "holonomy", which will be discussed in the following chapter. Definition 3.4. An Ehresmann geometry (P, 7) on M with group G, consists of a right principal G-bundle G >• P —?-*• M, together with an Ehresmann connection i:TP >g. 24 If (P, 7) is an Ehresmann geometry on M with group G, we have the following . short exact sequence 0 ^G-^P^-^M >-0, which induces the tangent short exact sequence 0 *- TpPG -^-*. TPP -^ TXM ^ 0, which shows that the vector fields on G, are all vertical, that is, they lie in ker(-7r*). In particular X^ € ker(7r*) for all X € g. Given any principal bundle, there is always, at least one Ehresmann connection on it. This will be shown the following two results. Lemma 3.1. Every point of M has a neighborhood U such that every connection defined on a closed subset contained in U can be extended to a connection defined over U. Proof. Given a point in M, it will be sufficient to prove this result for a coordinate neighborhood U of M with 7r_1(t/) = U x G, thus we consider the the trivial bundle U x G. A connection form is 7 is now completely determined by its behavior on the points of U x {e} where e € G is the identity element since R^ — Ad(h~1)'y. Let s : U *- U x G be the cross-section given by s(x) = (x,e), then 7 is completely determined by the g valued one-form 5*7 on U. Using the properties of the Ehresmann connection we may write any vector X E TS^(U x G) uniquely as X = Y + Z, where Y is tangent to U x {e} and Z is vertical so that Y — s*(n*X). We therefore have 7(X) = 7(S,M)) + 7(2) = (A)(a)+A where A is the unique vector in i) such that the fundamental vector field A* corre sponding to A is equal to Z at s(x). Since A is only dependent on Z, 7 is completely determined by s*j. The above equation shows conversely, that any given f) valued one-form on U uniquely determines a connection form on U x G. 25 Thus, in light of the above, it is enough to prove the result for g valued one-forms on U. Furthermore, if {A} is a basis for g, then UJ = ujlAi for usual one-forms u\. So again this extension problem is reduced to the usual one-forms on the coordinate neighborhood U of M. Let x1,... ,xn be a coordinate system on U. Then every one form on U is written as fcdx1, where the /;'s are functions on U. The result now follows from the fact that, a differentiable function on a closed subset of R™, can be extended to a differentiable function on all of R". D Theorem 3.1. Let G • *- P —^-»- M be a principal G-bundle, and A a closed subset of M. If M is paracompact, every connection defined over A can be extended to a connection in P. In particular, P admits a connection if M is paracompact. Proof Let / be some finite index set and {Ui}i€l a locally finite covering of M such that each Ut has the property of the previous lemma. Let {V^} be an open refinement of {Ui} such that V{ C Ui for each cl(Vi) C Ui for each i G I. Now for each subset J of J, set Sj — UjGjcl(Vi). Define T = {(r,J) : J G I, and r is a connection over Sj such that T\ADSJ — &} > where a is a given connection on A D Sj. Now introduce a partial order -< on T by defining (r', J') -< (r", J") if J' C J" and r' = r" on Sj. Then given any chain in T it is clearly bounded, so by Zorn's Lemma there exists a maximal element (r, J) of T. But then we have J — I and so r is a connection fulfilling our requirements. • Thus, as a consequence we have that every principal bundle G »- P n > M gives rise to an Ehresmann geometry (P,j). A more general concept than an Ehresmann connection, is that of pseudotensorial forms, which is a generalization of a tensorial form. Definition 3.5. Let V be an G module via the representation p : G >• Gl(H). A V valued r-form l R*g The set of all pseudotensorial r-forms is denoted by T(f\r T*P, V))G. Similarly, (Xu...,Xr) = 0 whenever at least one of the Xi 's is vertical. The set of all tensorial r-forms is denoted r byrhor(/\ T*p,vf. Example 3.1. Condition (1) for an Ehresmann connection implies that 7 is pseu dotensorial of type (g, Ad), and since the fundamental vector fields are vertical, con dition (2) says that it is not tensorial. 3.1.2 Cartan Connections The notion of an Ehresmann connection is very general, in the sense that every prin cipal bundle is equipped with at least one. Therefore, these Ehresmann geometries, although interesting, would seem to be not as interesting as Cartan geometries. A Cartan geometry consists of a principal bundle equipped with a Cartan connection. This section of the chapter is going to be used to determine necessary and sufficient conditions for a Cartan connection to exist on a principal bundle. Definition 3.6. Let H *- P —^-»- M be a principal right H-bundle. A Cartan connection on P, is a pseudotensorial one-form UJ, of type (g, Ad) for some Lie algebra g satisfying I) < g, such that UJ defines an isomorphism TPP —-*- g for each p € P, and UJ reproduces the infinitesimal generators of the fundamental vector fields on H. Writing more succinctly we have 1. UJ : TpP *• g is a linear isomorphism for all p € P 1 2. R*huj = Adih- )^ for all h 6 H 3. u(Xi) = XforallXet)- Condition (1) says dim(P) = dim(g), and that keroj = {0}. Thus the Cartan connection does not give rise to a notion of horizontality. However with the dimen sional condition, we have that there is a stronger relationship between P and g, this 27 condition will allow us to generalize the usual linear tangent space to a homogeneous space and to "roll" this tangent homogeneous space along M. Condition (2) implies that a; varies smoothly over the bundle, and (3) is equivalent to saying: UJ takes values in the sub-algebra rj < g on the vertical vectors, and in fact restricts to the Maurer- Cartan form UJH '• TPX >• f) on the fibres of P. These conditions will become more important when we introduce the notion of a Cartan Geometry. The curvature of the Cartan connection cu is given by the familiar formula tt[u] = du + -[cu,a>]. The connection u) is said to be flat if Q[UJ\ = 0. In addition to the curvature two form £1 we have what is called the torsion, represented by T. The torsion of UJ is the projection of Q[UJ] onto g/h. To clarify, we have the following diagram T A\P)—* ^g ^a/f, Since H C G only implies that f) is a subalgebra of g, the quotient space g/Fj is merely a vector space. For this space to be again, another Lie algebra, rj would have to be an ideal in g. That is \) would have to satisfy the extra condition [h,g] C h. Example 3.2. The simplest example of a Cartan connection is the Maurer-Cartan form u>a- Let H and G be Lie groups such that H C G, and cl(H) = H, and consider the principal bundle H 5- G »• G/H. By the properties of the Maurer-Cartan form, it is G-equivariant, and hence H equivariant, and UJQ reproduces the infinites imal generators of the fundamental vector fields on G, by definition, and so it does on H as well. The Lie algebra property is also satisfied immediately. Furthermore, UIG satisfies the Maurer-Cartan equation, duo — —1/2[U>G,CJG], which shows that the Maurer Cartan form is a flat Cartan connection. The curvature Q, of the connection u>, is closely related to the curvature of P. The curvature of P, denoted by K, can be computed from Q as K(X, Y) = 1 1 Qp(uj~ (X),uj~ (Y)). The curvature K may be interpreted as a section, called the curvature section, of the associated vector bundle P x# Hom(A2(g/h),g), where the action of H on Hom(g/h,g) is given by h • ip'= Ad(/i)^(A2(Ad(/i-1)). 28 Definition 3.7. Let u> be a Cartan connection on H >- P v > M, ui is said to be reductive if there is a Ad(H)-invariant subspace p of g such that g = f) © p. Furthermore, u is said to be first order reductive, if the adjoint representation of H is faithful (i.e. infective). Remark 3.1. The requirement of p being Ad(h)-invariant is equivalent to saying, that g has an H-module decomposition g — hffip. Also, if ui is a reductive connection, then it may be written as to — co^^cop, and hence the reasoning for ui being called reductive. Proposition 3.2. Let u> be a reductive Cartan connection on H 5- P —2-*. M with Lie algebra g. Then with u> = a^ © u>p, UJ^ is an Ehresmann connection. Proof. Immediately we have that u^ : TP -—->- h, and that LO^ is iJ-equivariant. The equivariance comes immediately from the equivariance of CJ, Rh*oj — Ad(/i_1)w. To see this, consider Rh*^ = Rh*(wt) + up) = Rh*ut) + Rh*up, from which it follows _1 _1 Rh*^t) + Rh*up = Ad(/i )a^ + Ad(/i )a;p, which implies _1 _1 Rh*u>t, - Ad(/i )w(, = Ad(/i )c<;p - Rh*uP- Since the left hand side lies in f) and the right hand side lies in p, with h D p = {0}, both sides vanish, hence Rh*Uf, = Ad(/i-1)^. Now let X E i), then by definition co(X^) = X from which it follows wfl^^ + ^Gf), hence u>p(X^) = 0, and thus LJ^{X^) = X for all X € h. Therefore u^ is an Ehresmann connection. • 29 The Maurer-Cartan form UG on H >• G * > M can be uniquely expressed as U)Q = u)lj If the g is reductive, we may replace * with a zero, and write this more compactly l as u>c = ^ij © u/p where ujf, — (co j), and u;p = (cu;), with p = g/h. In light of this discussion, we have that u>a is a Cartan connection, and in the last reductive Cartan connection. In addition to ui splitting up into a fj part and a p part for a reductive Cartan connection, which may be shown by the following diagram adapted from [9] TP- -9 we have the curvature also reduces Q,[UJ\ = fi^ © T, and a similar diagram follows A2(P)— *g where fij, is the curvature of the Ehresmann connection u;^ and T is the torsion of u. 3.1.3 Existence of Cctrtan Connections One of the conditions for a Cartan connection to exist on P, is that there exists a soldering form. In the context Riemannian geometry, this reduces to the notion of a coframe field on a given manifold. 30 Definition 3.8. Let M be an n-dimensional manifold, H >• P v > M and p : H >• GL(M.n) a representation. An M.n-valued tensorial one-form: 9 : TP -» Rn is called a soldering form. Example 3.3. Every H-structure P has a natural soldering form, which is called the fundamental form, defined by -1 ep{X) := p 7r„P0 VpEP andX E TP. p is given by the natural action of H < GL(M.n) onMJ1. Remark 3.2. The definition of the fundamental form relies on the fact that the tangent bundle TM is associated to P, or in other words, P is geometrizable. Definition 3.9. Let V E —^-*- M be the associated (vector) bundle of H *- P n > M. Elements ofT(/\rT*M,E) are called bundle valued r-forms. Lemma 3.2. Let P, V and E be as above. Then the following are in bijective corre spondence 1. bundle valued r-forms on M 2. tensorial V valued r-forms. Proof. ((1) '—> (2)) Let p : H »- GL(V) be a representation and E the bundle associated with P with fibre V on which H acts through p. Let 4> be a bundle valued r-form of type (V, p) and define 9S(x1,...,xr) = p(0(x1*,...,x;)), n where Xi E TXM, p £ P giving the isomorphism : R >•. Tn(p)M, with n(p) — x and X* is any vector at p such that 7r(X*) = Xj for i = 1,... ,r. We then have 0(Xi,..., X*) 6 V and so p : V -» 7r^1(x) is linear, and ((2) <-» (1)) Let 0(x*,..., x;) = p-\ipx{ Thus we have the natural identification H r I/\T*M,E J ^ rhor |/\T*P,T/) J. Lemma 3.3. If a principal bundle H -—>- P —^->- M has a first order reductive Cartan connection UJ = u^ © wp, i/ien P can 6e interpreted as a principal frame bundle with Ehresmann connection u>^ and soldering form uv. Proof. We know that u)^ is an Ehresmann connection from before, and the fact that ujp is a soldering form follows from the definition. Since u is first order reductive, ker(Ad) = {0}, and hence we can identify H with a subgroup of GL„(R) (n — dim(p)). Let p 6 P lie over x G M, then we have the short exact sequence 0 — Tp(pH) -^ TPP -^U- TXM — 0, along with another short exact sequence, 0 ^-Ug-^j/d. -0. Now using the properties of the Cartan connection we may combine these to produce an isomorphism of short exact sequences n*p 0—+Tp(pH)^TpP^TxM—~0, U1H b- "^0 Q/i) where [}P~\ei),..., ¥>~ ^e,,)) • h = (^(ei),..., We claim that the map : P *- Q defined by P'->(Pp1(ei)>--->Vp1(en))> is a bundle isomorphism. If X e TXM, then we may write X — TT*p(X) = TV*ph(Rh*X) for some X E TPP, consequently we have = p(ujph(Rh*)X) x = p{Ad(hr )wp(X)) l = Ad(h- )p{up(X)) x = Ad{h- )^p{^p{X)) l = Ad{h~ ) We are now ready to prove the main result, originally found in [4]. Theorem 3.2. Let H 5- p * > M be a principal bundle, then the following statements are equivalent: 1. P has a Cartan connection for some g > b 2. P has a reductive Cartan connection for some g > h 3. P has a reductive Cartan connection for some g > f) with [p, Jj] = 0 4- P has a soldering from 33 5. P is a geometrizable bundle. Proof. (3) => (2) =*• (1) is obvious. (5) =4> (4): Suppose P is geometrizable, then TM is associated to P via the H- module structure of Rn. Consider the identity section id G T*M n H r (f\T*M,TM\ Z* rhor I j\T*P,R ) ) then gives us a soldering form for P. (4) =4> (3): Let 8 be a soldering form. By Theorem 3.1 there exist an Ehresmann connection 7. With p := M.n viewed as an if module via the representation given in the definition of a soldering form, and g := f) © p. Then we have that [p,p] = 0, and we claim that UJ = 7 © 9, u is a reductive Cartan connection. Let X € h, then X* is vertical, and since 0 is tensorial by definition, 6{X^) — 0. Noting that 7(X^) = X, we conclude that u>(X^) = X. Moreover, since dim(M) = dim(p) = n, it follows that dim(TpP) = dim(fl), and hence uip : TPP -—»- g is an isomorphism (since 6 is surjective). By definition of the soldering form, we have that P can be a viewed as a reduction of the frame bundle associated to TM. Write 0 — (91,..., O71)1. Since 6 is surjective, for X € TPP, we can write TT,P(X) = e\X)eh where e = (ei,. .., en) is a basis for TXM, with 7r(p) = x, and the action of H on P is given by Now we have the relations 34 i d {X)ei = ir*p(X) = (TTO Rh)mP)X = ^*Ph{Rh*x) = ^(R^X^ei-h) = •&(RhmX)tijei l thus e\X) = O^R^X^j and hence R*h0 = h~ 6. Therefore = Kd{h-l)-y + M{h-l)9 = Ad(h-l)u. So with uj = 7 © #, we have that wisa reductive Cartan connection. (1) =4- (5): Let wbea Cartan connection on H *• P ——>• M. We will proceed to define a smooth bundle map q : P x g >- TM which depends on ipp. Using the map 0 *Tp(pH)^TpP^TxM -0 WH VP 0 ^h — 0 ^-fl/fc ^0 define the smooth bundle map Pxg T(M), q- (P,W)^ (7T(p), ¥>/(/!>(«/))) and note that q{ph,Ad{h-')w) = (T^^MM^M)) = {ir{p),{Ad{h)vph)-\p{w))) 35 So q induces a canonical smooth bundle map q : P x# g/f) >• T'M and since this is an isomorphism on fibers and induces the identity map on M, q is a vector bundle isomorphism, therefore TM is associated to P x# g/t), and hence H >- P *• M is geometrizable. D Using theorem 1.2.1, given an principal if-bundle with a Cartan connection, there exist a reductive Cartan connection, in particular, a first order reductive Cartan connection. A simple consequence of the above lemma, and last proof is now given. Corollary 3.1. The following are in bijective correspondence: principal H-bundles with first order reductive Cartan connections u — ui^ © up I principal H frame bundles P over M with soldering form up and Ehresmann connection UJ^ on P Proof. —*• is given by the last lemma. The converse if given by (4) =£• (3) in the previous theorem. • Definition 3.10. Let H -—*- P * > M be a principal bundle, and g a Lie algebra satisfying h < g. Assume that dim(TpP) = dim(g,) for all p € P. A generalized Cartan connection on P, is a Q-valued one-form, u : TP >• Q, such that 1. Ryu = yld(/i_1)tc7 for every h € H 2. uJ(Xt) = X for every lef). As we can see, the only difference between generalized Cartan connections and Cartan connections, is that we weaken the condition that ZJP : TVP *• g is an isomorphism, to the condition that dim(TpP) = dim(g), that is we still assume TPP = g, but not via ID. This concept will become particularly useful in the next chapter. 36 3.2 Cartan Geometry As mentioned at the beginning of last section, every principal bundle can be made into an Ehresmann geometry (P, 7), since there exists an Ehresmann connection on every principal bundle. The purpose of this section is to determine why a Cartan connection is "nice" to have on a given bundle. Before doing so, we introduce some prerequisite material. Definition 3.11. A Klein geometry is a pair of Lie groups (G,H) with H C G a closed subgroup, such that G/H is connected. For the purpose of Cartan geometry, it is useful to regard a Klein geometry as the principal iJ-bundle H *- G w > G/H together with the Maurer-Cartan form bjQ : TgG *• g. This is a principal i7-bundle since the fibre are the left cosets of H in G, and these are isomorphic to H as right H-sets. Since the homogeneous space G/H should not have a preferred base point, where in this case H is the preferred base point, it could be more useful to define a Klein geometry as a principal i^-bundle H »- P —^» M that is isomorphic to the principal iJ-bundle H >• G ^^ G/H: Definition 3,12. A Cartan geometry {P,OJ) on M modeled on the Klein geometry (G, H) with group H, consists of a principal H-bundle H >• P —^ M, together with a Cartan connection UJ : TP »- 9. The goal of Cartan geometry is to generalize Riemannian geometry by replacing the linear tangent spaces, with homogeneous spaces. Thus we would prefer to have an obvious basepoint in the Klein geometry, this basepoint being H, will serve as the point of tangency to the manifold M. This tangent Klein geometry serves to approximate the local geometry of M. 37 To see this, consider the case where M is a deformation of the sphere S2 embedded in Rn. We may thus represent this as an SO(2)-bundle SO(2) *- P —^ M. Since M is a deformation of S2, we wish to choose a tangent Klein geometry which represents S2. For this we have the principal SO(2)-bundle SO(2) -SO(3) -SO(3)/SO(2)^§2. Now consider the Cartan geometry (P,co) on M, modeled on the Klein geometry (SO(3),SO(2)). The conditions on the Cartan connection UJ, allows one to roll the sphere over M without slipping or twisting, namely the property that states that u> : TPP >• g is an isomorphism for each p E P. With every movement of the sphere on M consisting of a rotations and a transvection of the point of tangency, the isomorphism is stating that there is a unique way in which to match the combination of a rotation and transvection of the point of tangency. This is enough to say, §2 is rolling along M without slipping or twisting. This is the usefulness of having a principal bundle equipped with a Cartan connection. A useful way to describe Cartan geometry is to consider the following induced sequence of bundles H -G- -G/H P il_^pXi/Gf ^PxHG/H M where i is the canonical inclusion map, and if is the canonical inclusion map of H- bundles pw \p,e], with e € G the identity. The associated bundle TX : PXJJG/H >• •M, can be referred to as the bundle of tangent Klein geometries. This is due to the fact that it describes a bundle over M whose fibers are copies of the homogeneous space G/H, each with a natural point of tangency, that being H. This can be made explicit as follows: Let x G M, then the tangent Klein geometry at x is the fiber 7if-1(:r), and the point of tangency, with respect to this bundle, is given by the equivalence class 38 \p, H], where p € Px is any point in the fibre of the original bundle H >• P * > M. Furthermore, since \ph, H) — [p, hH] = \p,H], for h € H, this is well defined. Remark 3.3. Note that in the above diagram, the bundle G/H *• P XfjG/H *- M is not necessarily a principal bundle. For G/H to be a Lie group we require H to be a normal subgroup of G, H < G. In addition to Cartan connections, we have generalized Cartan connections, thus we have the analogous definition. Definition 3.13. A generalized Cartan geometry (P,cJ) on M, modeled on the Klein pair (G, H), consists of a right principal H-bundle H > P —^-*- M, together with a generalized Cartan connection U : TP >- g. As we can see, that once given a principal bundle, the connection completely determines the geometry. Chapter 4 Ehresmann Connections versus Cartan Connections Note on References: The main source of reference for the notion of holonomy in an Ehresmann geometry is [7]. Cartan holonomy, and it's connection with Ehresmann holonomy can be found in [8] and [9]. Material concerning the bijection between generalized Cartan connections and Ehresmann connections, the exterior covariant derivative, and the relation on the corresponding curvature two-forms, can be found in [1], [2J, and [3]. 4.1 Parallel Displacement From the study of Riemannian geometry, we have the notion of parallel displacement with respect to the Levi-Civita connection. However, since Riemannian geometry, in the language of Cartan geometry, is a torsion free Cartan geometry modeled on Euclidean space, the Cartan connection decomposes as the Levi-Civita connection and a co-frame field. So for a reductive Cartan geometry, one can explain the notion of parallel displacement by use of its version of the Levi-Civita connection, which has been denoted thus far, by u^. This parallel displacement takes place in the bundle of tangent Klein geometries, and is not particularly interesting since, u^ takes values in rj, and H itself is the stabilizer of the point of tangency, of the corresponding tangent Klein geometry. To describe parallel displacement in a general Cartan geometry, that does not fix the point of tangency, we need a relationship between Cartan geometries and Ehresmann geometries. 4.1.1 Ehresmann Parallel Displacement Let I = [t0, ti] C R, and a : [t0, t\] ^ Q be a map with a(t0) = q0 and a(ti) = qi, we will denote such a mapping as a : I >• (Q,qi,q2) or a : I *• (Q,q) if a(t0) = a(ti) = q. Let (Q, 7) be an Ehresmann geometry with group G, and recall 39 40 the definition of the horizontal subspace HQ C TQ, it is the set determined by the kernel of the connection 7, that is HQ = ker(7). A vector X G TQ is called horizontal if X G HQ. From the tangent short exact sequence 0 ^Tq{qH)-^+TqQ^+TxM. -0, we have that 7r* induces a linear isomorphism ft : HqQ ^ TXM, for each q G Q, and hence given any vector X € TXM, there exists a horizontal lift X G TqQ, with ir(q) = a;. As a consequence of there existing a unique horizontal vector in Q, for every vector in M, if qo G Q lies over x G M, and we are given any path a : \t\,t2\ *• (M, x, y) there exists a unique horizontal lift of a, a : I >• (Q, qo, q\), starting at qo G Q. To summarize, we have the following definition. Definition 4.1. Let (Q,j) be an Ehresmann geometry with group H and Lie algebra h. A path a : / >• Q is called horizontal if o~*j = 0, that is, if all the tangent vectors of a are horizontal. A horizontal lift of a path a : I *• (M, x, y), is a path a : I >- (Q,qi,q2) which is horizontal, that is, if a*j = 0. Let a : I >- (M,x,y) be a piecewise smooth path on M, and a : / >• (Q,qo,Qi) its unique horizontal lift. By varying qo in the fibre ir~l(x), we obtain -1 _1 a mapping from 7r (:r) onto the fibre 7r (y). With this mapping we get that q0 maps to q\, which is an element of H. The element of H, that gives the mapping h : l TT~1(X) >- ir~ (y), is called the parallel displacement along the curve a. Furthermore since every horizontal path, is mapped into a horizontal path by right translation, we have that h G H defines an isomorphism of the fibres n~1(x) and n~1(y). Parallel transport is simply the manner in which we arrive at 7r_1(y) to ir~1(x), and completely determines the parallel displacement. 4.1.2 Cartan Parallel Displacement To discuss Cartan parallel displacement, we first give some results to simplify the upcoming discussion. In addition to simplifying this notion, these results are of great importance for this chapter 41 Theorem 4.1. The generalized Carian geometries {P,uj) on M modeled on (G,H), correspond canonically and bijectively to the Ehresmann geometries (Q, 7), where G *~ Q = P XH G >• M is the associated bundle of P. Proof. By the end remark from the previous chapter, to determine this correspon dence, we must show that there is a bijection between the sets of generalized Cartan connections on H *- P n > M, and the Ehresmann connections on H ' > For Y € 0 the fundamental vector field on Q is given by *Q{?(«> 9)) = R*(q*(°u, 0g),Y) = q*(u,g)(Ou, Lg*Y). For a generalized Cartan connection on P, define the map ip as (cpcJ):TQ -g ( _1 1 ipuj(q*(u,g)(Rh*Xu, Lh-uLg*Y)) = r + Ad((/i ^)- )o;u(i?/l»X„) = y + Ad(^-1)Ad(/i)Ad(/i-1)a7„(X„) we have that (Xu,Yg) = (X^(u), -Rg*X) for some X e f), so that 1 (^(^(M),-^.*) = ( = -Adig-^X + Ad{g-l)uu(Xi(u)) = 0. To check the G-equivariance of (tpu>)(Rk*q*(u,g)(Xu, LgtY)) = ((pO(q*(u,gk)(Xu, Rk*Lg*Y)) = (gk)(Xu, LWk,Ad(fe )y)) = Ad^-^y + Ad^-1^-1)^^) -1 = Ad(& )(<£aJ)<7»(„)5)(Xu,Lff*Y)), 42 and since ( 1 ((f~ j)(Xu) = j(q In fact, we may'say more. Suppose (P,oJ) is a generalized Cartan geometry on M modeled on (G,H). Consider the corresponding Ehresmann geometry ((5,7), where Q represents the principal G bundle, G >• Q = P xH G *- M. If • Q is an H bundle map, then 7 may be pulled back by cp to form a generalized Cartan connection on P, that is, 1 : P- ^PxHG P *-+ [p,e], where e G G is the identity, will suffice for such a map Proposition 4.1. Let H »• P n > M be a principal H-bundle, and G »- Q = P[G] = P Xfj G >• M its associated G-bundle. Then there is a canonical isomorphism of fibre bundles 43 PxHG/H ^QxGG/H M Proof. Let t : P *- Q be the canonical inclusion map sending p i-> \p,e\. Then, this induces a canonical inclusion map of associated bundles i! : P xH G/H —- Q xG G/H [p,gH}< -Hp),gH}. Since any element of Q xGG/H — P X^G XQG/H is an equivalence class \p,g',gH], with p E P, g' £ G, and gH G G/H, any such element can be written as \p,e,g'gH]. Therefore, we can define a map <;:QxGG/H *PxHG/H \p, 9',gH] i [p, g'gH]. This is easily seen to be a well defined bundle map. Moreover, ^'b>3#l = s[i(p),pH] = q\p,e,gH] = \p,gH], and t'<;\p,g',gH} = t'\p,g'gH}=[t(p),g'gH] = \p,e,g'gH] = [g,g',gH], hence q = c'"1 is a bundle isomorphism. • With this canonical bundle isomorphism, and the bijection between Cartan ge ometries and Ehresmann geometries, we can now describe parallel transport in a general Cartan geometry. Let (P,u;) be a Cartan geometry on M modeled on (G,H), and ((2,7), its asso ciated Ehresmann geometry with group G. Let a : [ti,t2] -—*~ M be a path on M, and [p, gH] be an element of the tangent Klein geometry at x = cr(t0) e M. Then 44 via the isomorphism given above, we may think of x as a point inQxc G/H. Then by using the Ehresmann connection on Q — P x# G, we may translate along a, then send the result back into the bundle of tangent Klein geometries, P x# G/H, with <;. That is, the parallel transport is given by where a : [£i, £2] *• Q, lS the horizontal lift of a starting at t(p), with respect to the connection 7. Thus, as one can see, this notion of parallel displacement does not fix the point of tangency, and thus allows one to roll a geometry on another. 4.2 Holonomy In addition of parallel displacement, there is the notion of holonomy, which can be interpreted in the terms of parallel displacement., 4.2.1 Ehresmann Holonomy Let a : I >- (M, x) be a loop, where x lies under the point q € Q. The curvature Q[^y] measures the failure of closure of the horizontal lift a : I *• (Q,qo,qi) in Q. A similar concept is holonomy, for this we have the following definition. Definition 4.2. Suppose that (Q, 7) is an Ehresmann geometry with group H, and a : I 9- (M, xo) is a loop on M. Fix qo G Q lying over xQ, and let a : I *• (Q,qo,qi) be the unique horizontal lift of a starting at go- The element h E H such that qo — qi- h is called the Ehresmann holonomy of a with respect to qo. By noting that qo and gi lie in the same fibre over x0, and since H trivially acts transitively on itself, there does indeed exist such an h £ H. One may notice by the interpretation of-ft [7], and the notion of holonomy, they seem very similar. They are in fact related, and this result is known as the Ambrose- Singer theorem, which will not be discussed here. Proposition 4.2. Let (Q,i) be an Ehresmann geometry and a : I ——*• (M,x0) be a loop on M. Fix qo G Q lying over XQ, if the holonomy of a with respect to qo is g, _1 then the holonomy of a with respect q0 • h is h gh. 45 Proof. Let a : I >- (Q, q0, q{) be the horizontal lift of a. Then since (Rho&y l qo-h = (q1-g)-h= (qx • h) • {h~ gh), 1 and hence the holonomy of o~ with respect to q0 • h is given by h~ gh. • The three important properties of Ehresmann holonomy in which will be in com parison with Cartan holonomy are the following: 1. Has values in the fibre H 2. The holonomy of each a curve with respect to different points in Q varies by conjugacy according to the starting point of the lift 3. Every loop on M has holonomy. 4.2.2 Cartan Holonomy Let (P,w) be a reductive Cartan geometry on M, modeled on the Klein geometry (G, H). Then the Lie algebra g decomposes as g = f) © p for some Ad(#)-invariant subspace p of g. Consequently the Cartan connection ui decomposes as ui — Wf, © u>p. As we saw earlier, the connection o^ is an Ehresmann connection, and hence defines a horizontal subspace HP of the tangent bundle of P such that TP — VP © HP, where VP = ker(7r*) and HP = ker(o;(,). However, if (P,u>) is not reductive, then we loose the notion of a horizontal lift, since u> does not have a kernel. As in the case for Ehresmann connections on principal bundles, we define a horizontal lift of a vector X G TM as the unique vector X £ TP such that TT*P(X) = X^p) for all p £ P and X G HPP, and the horizontal lift of a path a : I »- (M,x,y) to be the curve a : I >• (P,p,q) such that every tangent vector lies in the horizontal subspace of TP, i.e. a*Uf, = 0. 46 Definition 4.3. Let (P,w) be Cartan geometry on M modeled on (G,H). Given a piecewise smooth path on P, a : I *• (P,p,q), the map a : (I, to) *- (G,g) satisfying a*(cuG) — a*(uj) is called the development of a. Before discussing the holonomy of a Cartan geometry we must know that we can extend the notion of development of paths on P, to that of developments on M. In contrast to paths on P, which develop on G, a path on M develops to a path on the model space G/H. To see this we have the following lemma and proposition. Lemma 4.1. Let (P,ui) be a Cartan geometry modeled on (G,H). Let (a:I *(P,p,q) \h:I *H be two piecewise smooth maps. Then (a • h) = ah, where the development (a • h) starts at h(to) and a starts at e € G. Proof. First we note that (ah)(tQ) — h(t0) = a(to)h(to), which shows that both developments start at the same point. Using the formula to compute the change of the Darboux derivative, we have that since (ah) G H 1 (ah)*ujG = Ad(h~ )au>G + h*u>G- To compute (ah) UJG we note that by definition of development: (ah) uiG = (ah)*ui. By writing the product of ah as the composition I ——»-1 x I >• P x H *- P 11 (t, t) i (a(t), h(t)) i ^ a(t) • h(t), we obtain 47 (ah) ijjG = (ah)*u = (/i o (a x h) o A)*w = ((ffXft)oA)*|i'u 1 = ((ax /i)oA)*(Ad(/i- )cr*a; + /i*a;i?) 1 = ((or xh)o A)*(Ad(/i- )7r>a; + TT^CU^) 1 = Ad(/i- )(7rJ o (o- x /i) o A)*u + (TT*H O (a x h) o A)*wH = Ad(/i_1)cr*a; +/I*O;H 1 = Ad(/r )<7*u;G + /i*u;G. From the uniqueness of the Darboux derivative, the result now follows. • Proposition 4.3. Let (P,u>) be a Cartan geometry on M modeled in (G,H). Let a : I *- (M,x,y) be a piecewise smooth path, a : I >- (P,x,y) be any lift, and a : (I, to) *- (G, e) be its development on G. Then its image a = iro : (I, t0) >• (G/H,e) in G/H is a curve that is independent of the choice of lift-a. Since for each vector X E TM there exists a vector X E TP such that ir*X — X, we have that for any curve on M there is always a lift to a curve on P. Thus we have the following definition. Definition 4.4. Let (P,UJ) be a Cartan geometry on M, modeled on (G,H). If a : / *- (M,x,y) is a path on M, the development of a is the development of any lift on G to P of a projected onto G/H. Intuitively, one can think of the development of a path a on M, as the path traced out by the points of tangency on G/H, from moving G/H along a. For a reductive Cartan geometry, with connection u = u^ (BUJP, the curvature fi[o;^], is modified in such a way that the model geometry becomes the standard for flatness. That is, a horizontal lift of a loop on M, is again a loop on P, if f^u^] = 0. Let 7Ti(P,p) and TTI(M,X) denote the fundamental groups of P and M at p and x respectively. Unlike the case with the Ehresmann connection, where every loop has a well-defined holonomy, only the loops on M whose classes lie in the image 7r* : TVi(P,p) *~ TTI(M,X) have well-defined holonomies. That is since the holonomy only applies to loops on M which lift to loops on P. Note that in the case with Ehresmann connections, holonomy applies to every loop on M. 48 Definition 4.5. Let (P,w) be a Cartan geometry on M modeled on (G,H). Fix p £ P such that 7r(p) = x, let A : I >• (M, x) be an element of 7m(7r*) (thus Im(X) is an equivalence class of path homotopic mappings denoted [X]), A : / *• (P,p) be any lift (a lift of any path in the path class[X]) , and X its development on G. Then X{t\) is called the holonomy of the loop X with respect to p. The holonomy of (P,u>) with respect to p is the set <&(p) C G of all holonomies. The last result about Cartan holonomy is now provable. Proposition 4.4. Let (P,UJ) be a Cartan geometry on M, modeled on (G,H), then the following holds: 1. $(p) is a subgroup of G 2. if the holonomy of the loop X with respect p is g, then its holonomy with respect to ph is h~lgh. In particular Proof. (1). This is clear. (2). Let A : / >• (P,p) be a loop on P, and consider the loop Rh,o X : I -—>• (P,p • h). Then since 1 (Rh o X)*LU = X*R*hoo = X*Ad(h- )u = (RH7\yuG, it follows that the development of Rh o X is given by C(/i-1)A. Therefore, since the holonomy is given by the end point of the development, we have that {C{h~l) o A)(tij = Cih-^Xfa) = C{hTl)g = hTlgh. The result that $(p • h) = h~l${p)h now follows from here. • With a subset p c /ra(7r*), the holonomies corresponding to classes in p are called restricted holonomy groups, denoted by $p(p), and form a subgroup of the holonomy group $(p). It is easy to notice that Cartan holonomy is much more complicated than that of Ehresmann. Some the properties of Cartan holonomy that we have seen are as follows 49 1. H does not necessarily contain the entire holonomy group, since $ C G, not H. 2. The holonomy varies by conjugacy according to the starting point of the lift. 3. Holonomy applies only to loops on M which lift to loops on P. As we can see, only property (2) of Cartan holonomy coincides with that of Ehres- mann. However, the two types of connections can be associated to each other if there is an H bundle map cp : P >• Q such that the Ehresmann connection on Q satisfies a given property. When this property is satisfied, the holonomies turn out to be identical. 4.2.3 Ehresmann Holonomy as Cartan Holonomy Proposition 4.5. Let (P,ui) be a Cartan geometry on M modeled on (G,H), and let (Q, 7) be the Ehresmann geometry given by the correspondence above. Suppose x0 € M and the loop c : I *- (M, Xo) are fixed, and c lifts to a loop based at qo G P, then the Cartan and Ehresmann holonomies are equal. Proof. Given such a loop c on M, let c : / »- (P xH G, (pa, e), (p0, g)) be a horizontal lift of c, and let c = {ci,c2) : I (P x G, (p0,e), (po,g)) be a lift of c into P x G. The Ehresmann holonomy of c is clearly given by g~*. It may be shown that for a connection 7 on P x G, we have 7^) = Ad(<7-1)7rpCJ + 1 1 TTQWGI and X) •, — (u>~ (X)p, —u!~ (Ad(g)X)g), from which it follows that 1 1 1 1 W* ) - 7(P,9)(^ (*),-^ (Ad(9- )X)) 1 1 1 l = Ad(g- )TT*Pu^- (X),-^ (Ad(g- )X)) l 1 + n*aUjG(u;- (x), -^(AdGT )*)) = AdGr1)* - (AdQr1)^)) = 0. 50 Thus 0 1 = c*7(M) => 0 = Ad(^- )a*7r>w + ~C*ITGUJG 1 = Ad(g~ )(7TP OC)*LO + (irGoc)*uG l = kd(g~ )c\u + c*2uG 1 l therefore c\u — —Kd{c2)c*2ijOG = (c2 )*cuG. So by definition of development, c2 is l l the development of c on G, and thus the Cartan holonomy is given by c2 (t\) = g~ , showing that the two holonomies are equal. • Remark 4.1. The fact that c2 is the development of c on G, follows from that fact 1 l that C\ is a lift of c to P, and (C2 )*LOG = C\UJ, and hence the holonomy is g~ since l c2 {h) = g-\ It be important to note that for a reductive Cartan geometry (P,ui), u may be written as w = W(, ©wp, and since UJ^ defines an Ehresmann connection, one should consider if the Ehresmann holonomy has anything to do with the Cartan holonomy. For this we have the following theorem. Theorem 4.2. Let (P,u>) be a first order Reductive Cartan geometry on M modeled on (G, H) with G = H K L, and the action of H on L is given by the representation p. Fix XQ € M and a loop A : I *- (M,xo) that lifts to a loop bases at go € P. Then the Ehresmann holonomy of c is the H component of the Cartan holonomy. Proof. Let g,h, and p be the Lie algebras of G, H, and L respectively, and Q be the associated frame bundle as given in the previous chapter. Since we may choose a lift c\ : I *- (P,qo) of c, which is a closed loop, and there exists a unique horizontal lift c2 : I -—>- (Q, qo,qi), there exists a map h : (I, to) *- (H,e) such that C2(t) = Ci(t)h(t). Thus we have that q\ = c2(ti) = cx{tx)h{tx) = goM*i), and hence the Ehresmann holonomy is given by h~x{t\). Since G — H K L, we have that 0 = h © p and hence u> = u/j, @OJP. Thus by pulling back oj by c2 we obtain C^LO = C2U)f, + C2UJp = C*2UJp, 51 which implies C*2OJ : TP *- p C Q- From this, we see that the development of c, developed at(e,e) lies entirely in L, and thus ends at (e,l) for some / £ L. Let h(t) = h(ti — t), then the development of the curve q0h starting at (e,e) is given by l h{t\)h and thus develops to the curve starting at (e, I) is given by (e, l)h~ (tx)h. From here it follows that the development of (qoh) o ci starting at (e, e) has the endpoint, and hence Cartan holonomy {e,l)hr\tx)h{tx) = {hr\tx\P{h{tx)){]). Therefore the the H component of the Cartan holonomy is given by the Ehresmann holonomy. D 4.3 The Covariant Exterior Derivative The covariant exterior derivative is a very useful tool, in which many formulas can be simplified (e.g. curvature of a connection). Of particular interest (which will unfortunately only be mentioned and not explored), is the use in co-homology theories. The authors of [1], [2], and [3], explore this in more recent papers. For now, we have the following definition. Definition 4.6. Let (Q,-j) be an Ehresmann geometry, V a vector space with basis r {ex,..., en}, and ip € /\ T*Q®V be aV valued tensorialr-form. With Xx,... Xr+i € TQ, the covariant exterior derivative of (p with respect to j is defined as H dytpiXu.. .,Xr+1)) = dtpiX?,.. .,Xr +1), a where dip — d With this definition, we may redefine the curvature of a connection 7 on a principal bundle H »- Q * > M, as QE '•= ^77 € f\ T*P For a generalized Cartan connection (not necessarily reductive), ZJ, on H >• P —^-a- M and any representation p : g > GL(V) we define the exterior covariant derivative as r r+1 W i »- dty + p(uj) A * Recall the correspondence given earlier between H equivariant maps on H >• P ^ M, and global sections on G *- P[H] = PxHG *- M. We may find that the covariant exterior derivative of a reductive Cartan geometry is identical with the covariant exterior derivative in the associated Ehresmann geometry obtained from the above discussion. Let ip be the map giving the bijective correspondence, then the covariant exterior derivative with respect to a reductive Cartan geometry (P,u>) on M is given by the operator Dx : T(P[G}) r(P[G]) defined by i>(Dxf) = X(i;(f)), where X is the horizontal lift of the vector X. By this simple discussion it is seen that the two definitions are identical. We now have that the exterior covariant derivatives are identical when the ge ometry (P,UJ) is reductive, however, the natural question arises: What if (P,OJ) isn't reductive? To answer this question, we use the correspondence given between gener alized Cartan connections on P and Ehresmann connections on P[G]. But first, we need the following lemma. Lemma 4.2. Let V be a vector space and, (P[G], 7) be an Ehresmann geometry, where P{G] is the associated principal G bundle G »- P[G] 5- M to H —-*- P n > M. H Then if $ € Thor(T*P[G], V) , and p : G »- Gl(V) is a representation, d7$ = d$ + p'(j) A $, s le where p' — p*e '• d *~ sKV) ^ ^ differential of the representation. G Proof. Let $ E rhor(r*P[G], W) , and take a vertical vector field on P[G], say X\ the fundamental vector field to some X € g. If we insert this vector field into d70, 53 r r 1 we get 0 by definition. Now let ix • A (P[G]) >• A ~ (P[G}) denote the interior product defined by - ixu>{Xl,...>Xr-l):=u)( 1 3 12 l ix = -X ujji2„Ardx A • • • A dx (r-1)! 1 r = — 53 ^^i-'v-vC-iy-1^1 A • • • A dx'J A • • • A da*\ . Then we have i^t^ = 0 again by definition (since X^ is vertical), and by denoting the flow of X\ by at, we have d_ C ^ = x dt d_ p(exp(-tX))* dt -f/(X)$, hence iXi(d$ + p(j) A$.) A = ixtd$ + dixt + /o'(ixtw) A $ - ,c/(u;) (i*t$) = (ixtd$ + dixt) + /o'(ixtw) A$ = £*t$ + p'POA = -p'(X) A $ + /9'(X) A $ = 0. Since ixt$ = 0 => ®(Xi,..., X\ ..., Xr) = 0 for any vector valued differential r-form d^{YuY2) = (x*d^)(YltY2) = d*(YuY2), 54 and , (d* + p (7)A$)(y1,y2) = d$(y1,y2), since Yi,Y2 & ker(7). Therefore d7$ = d$ + p'(7) A $. Theorem 4.3. Let (P,tD) be a generalized Cartan geometry on M modeled on (G, H). Then there is a canonical linear isomorphism, H G K : Thor(/\T*P,d) ^rhor(/\T*P[G],Q) which intertwines the covariant exterior derivative of any generalized Cartan connec tion u> with values in Q and of is canonically associated Ehresmann connection tpuj on P[G}: p p+i H d^ o K = K o d* : Thor(/\ T*P, Q) - Thor(/\ T*P[G},gf. Proof. Define the map P p+i T p H * ••r hor( A * i Q) ^ rhor (f\ T*P[G] ,Qf. by 1 *(*)«»*)(*(& La. PxG P*HG projF P P/H A vector q*(£u, Lg(Y)) is vertical in P[G] if and only if it is of the form 55 for Z G g and X6l), thus it is horizontal. It is also seen that «;*& is well defined only if ^ is horizontal. H Now let $ G rhor(TP[G],g) , then and hence 1 1 (KK-i)$)g{u>9)(q*(eu,Lg*Y\...)) = M(3- )(K- *U&&...) = Ad^-W^^Oe),..-) = *(Rg.(q.(£,Oe),...)) = *(^(^,V0e),...) = *(q*(&Lg.Y),...), which follows since <& is horizontal, and similarly and thus KT1 is indeed the inverse of K. The proof showing that K$ is G-equivariant is similar to the proof of equivarience given in theorem 4.1. Therefore we have that K p H p G is an isomorphism: rh0T(/\ T*P,g) —'*• ThoT(/\ T*P[G},Q) . Now we must check the intertwining of the covariant exterior derivativewith respect to u> and KU>. First we compute d^ o K. Using the previous lemma we have that dvW$ = d<& + [ipuj,$], for $ € rhor(T*P, V). We begin by computing d^JS. So let £ G TP.be a Pf-equivariant vector field, and let Y? denote the fundamental vector field of Y* G Q. Then we have that & x ly is AT-equivariant, and yields a corresponding vector field & x r/ on rP[Gl = TP XTH JTG. We clarify these ideas with the following diagram PxG €'xrt >TPxTG P[G] ^ixyt , TP[G] that is #*(& x y/) = (j x v/. As a consequence we have 56 We may now proceed with the calculations. t d(K*) (6xF1 1...^1xd) P+I i x Y f =Y,(~y& <) u* 6 x n ...,;... {P+i x[ p+YIl i=l i+j X + J2(-l) (^)(kxYl^xYj ,6^V.. ) «>•••? J ) • • i Sp+1 *p+l ) > i (^)q(u,g) (^x~Yl...^p^x~Y^ = (K*),(U>9) (g*6(«). -^5*^2, • • • , ?*(£p+l(«), -kg^p+l)) = Ad^-1)^^^), • • • ,<£p+i(«)) € g we get 1 = AdOr^y,)*^,... ,4+1) + AdGr )^ &, • • -,M) 1 1 = - [Yt, Ad(5- )*„(6, • • • ,&>+i)] + Adfo- )^ (6,... ,^i)). Now inserting in to the formula for d(nty) we achieve d(K*) (6 x Yl.. . ,£p+1 x Y}+1)(q{u,g)) P+I 1 = ^(-i)*[y*,Ad((7- )^(6,...,^+i)] 1 +Ad(5- )(^)u(6,-..,4+i)- Furthermore f [cpCJ, K^f] 6 X Y},..., £p+i X yp +1 J (g(«, g)) p+1 i=l p+1 1 a ]T(-iy [y + Ador )^), Ad(p- )*u(ei, •..,&,... ,Ui) j=l p+1 1 £(-1)' [Fi; AdGT )^!, • • • ,&H-I)(«)] , i=l 57 and hence f (dp* o K)(U>9) (& xy/,..., £p+1 x Fp +1) 1 = Ad(^- )(d* + [a7,*])u(6,...^P+i). To finish, we compute (« o tfc^),^) Ui x Y"/,... ,^+i x FpVJ = AdGr1)^*)^!, • • • ,4+i) = Ad(g^)(d^ + p,^}u(^...,(p+1)), and thus we have the following commutative diagram p H p G rhor(A ^5) rhor(A P[GU) ) P+p+ 1 P „\H p+i G rhor(A ^0) •«— rhor(A P[G],g) ) i.e. G^G 0fi = K04 D Corollary 4.1. Le£ (F, w) be a generalized Cartan geometry on M modeled on (G, H).. Then there is a canonical linear isomorphism « : Thor(/\T*P,Vf *rhor(/\T*P[G],Vf, for any vector space V, which intertwines the covariant exterior derivative of any generalized Cartan connection u> with values in Q and of is canonically associated Ehresmann connection (pu> on P[G]: P p+i H T P d^ o K = « o db-: Thor(f\ T*P, V) Tftor( A * [G], Vf. Proof. Let p : G >• G1(F) be a finite dimensional representation, then we may rep resent the generalized Cartan connection as taking values in V, by p. The remainder of the proof is now exactly the same as above by using this representation. • 58 In addition to the intertwining property of the covariant exterior derivative, we have that the curvatures are related as well. Theorem 4.4. Let fi be the curvature of uj, then tpQ is the curvature of of cpZJ on P[H], where Proof. Q, € ThorCA P*PJ9)H) however cu is not horizontal, thus we modify the above calculations. Prom the definition of ip at the beginning of the section, we have = Yi + Adih-^U^i) from which we find 6 x y/ ( d{ \\ifxD, ipu] ^i x Yl 6 x Y^j (q(u, g)) 1 1 = ||r1 + Ad(5- )S7u(6),K2 + Ad(^- )S7u(6)] 1 -\\Y2 + Ad^Kfe),!! + Ad^- )^^)] 1 1 = fc,AdGr )!^)] - [Ti,Ad^" )^^)] + [Y1,Y2] -iAd(p-1)[aJ,a7](6,6)(^)- Adding the last two expressions, we arrive at Conclusion The topic of this thesis was chosen upon the obsessive reading of [8], and it is recom mended for anyone who has the desire to learn more about Cartan connections, and Cartan geometry in general. It may not be totally clear that all of geometry can be studied with connections on principal bundles, however Euclidean, Riemannian, and Klein geometry can easily be seen to be special cases of Cartan geometries. For ex ample, we may say that a Euclidean geometry is simply a Cartan geometry modeled on Euclidean space, that is, we take the Klein geometry to be (Euc„,0(n)). A Rie mannian geometry, is given by a torsion free Cartan geometry, modeled on Euclidean space, and of course a Klein geometry is a Cartan geometry modeled on itself. With these definitions, we may continue to construct conformal, projective, Lorentzian, and Mobius geometry, among any other types of geometries. For this the reader is referred to [8]. It should be observed that the definition of Riemannian geometry given above, does not include the concept of a non-degenerate metric. However, it can be shown that this type of geometry does indeed induce a metric on M, which is unique up to a scalar factor g G Euc„. Given a principal bundle, it is desired to find at least a Cartan connection, but more desirable is a first-order reductive connection. By the theorem concerning the existence of Cartan connections, if we have a soldering form (co-frame field, in the sense of Riemannian geometry), a Cartan connection, or the bundle is geometrizable, this is in fact guaranteed, the problem is finding one. This theorem seems to have a great wealth of information, and can probably be used to prove some very nice results; however, it is not known to the author, whether or not this has been taken advantage of. With the use of bijection between generalized Cartan geometries, and associated 59 60 Ehresmann geometries some important results can be simplified. This, along with the canonical bundle isomorphism P xH G/H •• ~ > Q xG G/H, between tangent Klein geometries, and "associated tangent Klein geometries" the notion a parallel displace ment is rather simple after knowing parallel transport in an Ehresmann geometry. This particular isomorphism, is a rather new result [9], and again, has not been taken full advantage of. The results about Cartan holonomy, can almost definitely be sim plified upon further exploration, without the use of developments; however, this is also unknown to the author, and will be of particular interest in the near future. Similarly with the covariant exterior derivative. It should be noted that Ehresmann geometry is an unknown term to the author, but seemed rather natural after having the concept of Klein and Cartan geometries. With this in mind, the diagram explaining the generalizations of Euclidean geometry in the introductory chapter, could perhaps be expanded as follows: Euclidean Generalize Klein Georitietr y Symmetry Group Geon netry 1 Add ;ure Curvature - liemg innian Generalize Car tan Geon aetry Tangent Space Geon netry Ehresmann Geometry where the final arrow can signify the connection taking values in the fibre, and in general having a kernel. This diagram holds true, since it has been shown that Ehresmann connections generalize Cartan connections (the bijective correspondence 61 between generalized Cartan geometries and Ehresmann geometries) and indeed if P is paracompact theorem 3.1 says that every bundle gives rise to at least one Ehresmann geometry; however, this diagram may be misleading with regards to the time in which the theories were developed (since Ehresmann connections were introduced before Cartan connections). Bibliography [1] Alekseevsky, D. V.; Milchor, P.W., Characteristic Classes for G-sturctures, Dif ferential Geometry and its Applications (1993). [2] Alekseevsky, D. V.; Milchor, P.W., Differential Geometry of Cartan Connections, arXiv:math.DG/9412232 vl, 1 Dec 1994. [3] Alekseevsky, D. V.; Milchor, P.W., Differential Geometry of g-Manifolds, Preprint ESI 7 (1993). [4] Barakat, M., The Existance of Cartan Connections and Geometrizable Bundles, arXiv:math.DG/0206136 v2, 19 Jul 2007. [5] Duistermaat, J.J.; Kolk, J.A.C., Lie Groups, Springer-Verlag Berlin Heidelberg 2000. [6] Husemoller, D., Fibre Bundles, Third Edition, Springer-Verlag (1994). [7] Kobayashi, Sh.; Nomizu, K., Foundations of Differential Geometry. Vol I., J. Wiley-Interscience (1969). [8] Sharpe,R.W., Differential Geometry: Cartan's Generalization of Klein's Erlan- gen Program. Springer-Verlag (1997). [9] Wise, D.K., Macdowll-Mansouri Gravity and Cartan Geometry, arXivrfr- qc/0611154vl, 30 Nov 2006. 62 V) then d^ip = Q.f\ip, where Q is the curvature 7. In particular D2 vanishes if the geometry (Q, 7) is flat. 52