GEOMETRY OF HORIZONTAL BUNDLES AND CONNECTIONS

A Dissertation by

Justin M. Ryan

Bachelor of Science, Wichita State University, 2009

Master of Science, Wichita State University, 2011

Submitted to the Department of Mathematics, Statistics, and Physics and the faculty of the Graduate School of Wichita State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy

May 2014 ⃝c Copyright 2014 by Justin M. Ryan

All Rights Reserved GEOMETRY OF HORIZONTAL BUNDLES AND CONNECTIONS

The following faculty members have examined the final copy of this dissertation for form and content, and recommend that it be accepted in partial fulfillment of the requirement for the degree of Doctor of Philosophy with a major in Mathematics.

Phillip E. Parker, Committee Chair

Jeffrey Hershfield, Committee Member

Thalia Jeffres, Committee Member

Kirk Lancaster, Committee Member

Mark Walsh, Committee Member

Accepted for the College of Liberal Arts and Sciences

Ron Matson, Interim Dean

Accepted for the Graduate School

Abu S. M. Masud, Interim Dean

iii DEDICATION

For Adelaide and Aydan. Never stop looking for answers.

iv To those devoid of imagination a blank place on the map is a useless waste; to others, the most valuable part. – Aldo Leopold

v ACKNOWLEDGEMENTS

I wish to thank Professor Phil Parker for introducing this problem to me, and for sharing your insight, advice, and ideas. More importantly, thank you for teaching me the value of a strong base of knowledge of the fundamentals; for teaching me to trust the intuition you’ve helped me develop; and for always holding me to the same high standards to which you hold yourself.

I also wish to thank Professors Thalia Jeffres and Mark Walsh. Thalia, you have known me since the beginning of my time in Wichita, and you’ve always been supportive of my work. You’ve also played an important role in my development as an educator as well as a student. I will always be grateful to you. Mark, thank you for helping me develop my geometric intuition, and for showing me the importance of always having an arsenal of examples at my disposal.

Thank you to my friends Kamielle Freeman, Everett Kropf, Patrick Rinker, Sam

Sahraei, Jimmy Shamas, and Nathan Thompson for all of the conversations, seminars, and long study sessions. To everyone on the third floor of Jabara, thank you for making this entire experience an enjoyable one.

Finally, I wish to thank my family. Mom and Dad, you have always supported me in my endeavors, no matter how crazy or impossible they may have seemed at first. I could never have done this without your love and support. Nick, you have always inspired me to try to be a better person, and to be more cognizant of my place in nature and the universe.

Addy, you are my hope for the future. Laura, thank you for always being in my corner, no matter what, and always being willing to talk. Grandma, thank you for the daily emails, weather updates, and news about home. Aydan, your smile always brightens my day. Roxan, thank you for being my best friend.

vi ABSTRACT

An Ehresmann connection on a fiber bundle π : E →→ M is defined by prescribing

a suitable horizontal subbundle H of the tangent bundle πT : TE →→ E. For a horizontal bundle to be suitable, it must have a property called horizontal path lifting. This property

ensures that the horizontal bundle determines a system of parallel transport between the

fibers of E.

The main result of this dissertation is a geometric characterization of the horizontal

bundles on E that have horizontal path lifting, and hence are connections. In particular, it is shown that a horizontal bundle has horizontal path lifting if and only if its horizontal spaces are bounded away from the vertical spaces, uniformly along fibers of E.

In order for a horizontal bundle to admit a system of parallel transport or have

holonomy, it must be a connection. However, certain other geometric properties that are

usually attributed to connections are actually properties of arbitrary horizontal bundles.

These properties are studied in the case when E is either a vector bundle or tangent bundle,

accordingly.

vii TABLE OF CONTENTS

Chapter Page

1 INTRODUCTION ...... 1

2 MANIFOLDS AND BUNDLES ...... 5

2.1 Fiber Bundles ...... 5 2.2 Special Bundles ...... 8

3 HORIZONTAL BUNDLES AND CONNECTIONS ...... 16

3.1 The Vertical Bundle ...... 16 3.2 Horizontal Bundles ...... 18 3.3 Connections and Parallel Transport ...... 21 3.4 Kinds of Connections ...... 27

4 LOCAL DESCRIPTION: WONG ANGLES AND CHRISTOFFEL FORMS . . . 29

4.1 Local Subbundles of TE ...... 29 4.2 Wong Angles in TvE ...... 30 4.3 Wong Angles Along Ep ...... 31 4.4 Local Christoffel Forms ...... 33 4.5 Horizontal Spaces Along a Lifted Path ...... 38

5 HORIZONTAL BUNDLES ON TM ...... 41

5.1 Vector Bundle Structures ...... 41 5.2 Second-Order Differential Equations ...... 42 5.3 Covariant Derivatives and Connection Geodesics ...... 43 5.4 Jacobi Fields ...... 47

6 UNIFORMLY VERTICALLY BOUNDED HORIZONTAL BUNDLES ...... 49

6.1 Uniform Vertical Boundedness ...... 49 6.2 HPL if and only if UVB ...... 51

7 CONCLUSION AND FUTURE DIRECTION ...... 56

REFERENCES ...... 57

viii LIST OF FIGURES

Figure Page

3.1 The horizontal foliation of a horizontal bundle on T R that is not a connection. . . . 26

α 4.1 The action of a Christoffel form Γv on a single vector Xp...... 34

4.2 Wong angles in a 2-dimensional slice of single fiber TvE...... 37

ix CHAPTER 1

INTRODUCTION

In 1854, Bernhard Riemann gave his inaugural lecture at G¨ottingen,entitled On hy- potheses that lie at the foundation of Geometry. Here he introduced a new notion of geometry that differed from Euclidean geometry. In particular, Riemann’s theory did not include a parallel postulate. He posited that geometry should not only be studied on Euclidean space, but also on more general spaces that locally resemble Euclidean space. These spaces are intuitively built by pasting together open subsets of Euclidean space in a specific way.

Thus the principal object of study became an n-fold extended quantity, or what is today called an n-dimensional manifold. At the time, these spaces were the most general setting to do calculus, hence also the most general setting to study geometry. Today these are still the most general spaces to do what might be called standard calculus, and play a prominent role in differential geometry, differential and algebraic topology, dynamical systems, high energy particle physics and gauge theory, and many other areas of mathematics and physics.

In order to study geometry on a manifold, one must be able to compare tangent vectors at different points of the manifold. Without a parallel postulate, there was no longer a natural way to do this. What was needed was a way to generalize the notion of parallel translation in a vector space to parallel transport along a manifold. During the century that followed Riemann’s lecture, Christoffel [9], Levi-Civita [23], Weyl [33], E.´ Cartan [8], and others tackled the problem from different points of view. The result was what Weyl [33] called a connection on the manifold.

A connection on a manifold, in this classical sense, is equivalent to a system of parallel transport of tangent vectors along the manifold. In other words, it describes how tangent vectors change with respect to infinitesimal changes in the manifold. This can also be

1 described in terms of a covariant derivative, or by a second-order differential equation on the manifold [13]. For the definitive history of connections before 1950, see [18].

In 1950, Charles Ehresmann [17] defined a class of manifolds called fiber bundles, independent of Whitney and Steenrod [18]. He then defined a connection on a fiber bundle

π : E →→ M to be a suitable splitting of the tangent bundle TE = H ⊕ V , where V is the natural vertical bundle.

The vertical bundle V is the subbundle of TE that consists of all vectors tangent to the fibers of E. The factor H is a complementary subbundle to V , called a horizontal bundle on E. Since V is natural, a connection is completely determined by specifying a suitable horizontal bundle H . The classical connections of Ehresmann’s predecessors are special examples of Ehresmann connections on the tangent bundle π : TM →→ M.

For a horizontal bundle H to be “suitable,” it must have a property called horizontal path lifting, or HPL. Ehresmann acknowledged that HPL is a nontrivial property by including it in his definition of a connection. He also recognized that including HPL in his definition was necessary to ensure that these objects actually do what their name claims: connect the

fibers of the bundle via parallel transport along the base manifold.

Indeed, parallel transport is defined using horizontal lifts of paths in M to E. Over any path γ in M, a horizontal bundle determines a line field on the pullback bundle γ∗E along that path. The integral curves of this line field are called the horizontal lifts of γ, and these integral curves foliate the pullback bundle. If each leaf of this horizontal foliation meets every fiber of γ∗E, then this process defines a diffeomorphism between the fibers of E along γ, called parallel transport along γ.

The key to this construction is the assumption that each leaf of the horizontal foliation of γ∗E meets every fiber of γ∗E →→ I. If this assumption holds for every path γ : I → M, then H has HPL, and the splitting TE = H ⊕ V defines a connection on the fiber bundle

π : E →→ M. It is then natural to ask what can go wrong. What properties must H have in

2 order to ensure that it has HPL? Which geometric properties of a connection actually rely on HPL, and which ones are actually properties of any horizontal bundle?

Despite the fundamental nature of these questions, it appears that until very recently complete answers could not be found in the extant literature. The first such was given by

Parker [27, v1] in 2011. He proved that a horizontal bundle on a tangent bundle TM →→ M determines a connection if and only if its horizontal spaces Hv are bounded away from the vertical spaces Vv, uniformly along tangent spaces TpM. He called an H with this property uniformly vertically bounded, or UVB, and showed that HPL is equivalent to UVB.

The main result of this paper is the extension of Parker’s result to horizontal bundles on arbitrary fiber bundles.

Theorem 6.2.1 A horizontal bundle H on a fiber bundle π : E →→ M has horizontal path lifting if and only if it is uniformly vertically bounded.

While the general idea of the proof is the same–reduce the problem to a system of first order differential equations–one must be more careful in implementing this reduction in the general

fiber bundle case.

The main tools used in the reduction are introduced in Chapter 4: Wong angles and local Christoffel forms. Neither of these are new inventions, but their application to this problem is new.

Wong angles are adapted from the angles used by Wong [34] to study the geometry of Grassmann manifolds. A horizontal bundle can be described as a section of a special kind of Grassmann bundle over the fiber bundle π : E →→ M; viz. Proposition 3.2.5. Thus their usage is completely natural.

Local Christoffel forms can be thought of as a lifted version of the classical Christoffel symbols, or connection coefficients, from the fibers of E to the total space of TE along the

fibers of E. This point of view is necessary because without a vector space structure on the

fibers of E, there are no connection coefficients. All algebraic operations must therefore be performed in the tangent spaces to the fibers; i.e., the vertical spaces.

3 The local Christoffel forms can be described as local sections of the first jet prolonga- tion J 1E →→ E over the fiber bundle π : E →→ M. A horizontal bundle H can be described as a global section of J 1E; viz. Proposition 4.4.3. Thus the usage of the local Christoffel forms is also completely natural.

The relationship between the local Christoffel forms and Wong angles is simple and elegant, and as far as I know, new. This relationship is developed and explored in Chapter

4. In Chapter 5, some classical geometric notions such as second-order different equations, covariant derivatives, geodesics, and totally geodesic submanifolds are described in terms of the local Christoffel forms. Finally, the equivalence of HPL and UVB is proved in Chapter

6.

4 CHAPTER 2

MANIFOLDS AND BUNDLES

In this chapter, certain fundamental properties and constructions of fiber bundles are reviewed, and notational conventions are introduced that will be used throughout the paper.

2.1 Fiber Bundles

A familiarity with smooth manifolds will be assumed throughout, where by smooth

we shall always mean C∞. As all manifolds considered will be smooth, we shall readily drop

this attributive and refer simply to manifolds. We write M n to denote a manifold M of

dimension n.

The main object of study in this dissertation is a smooth fiber bundle. These spaces

are built out of three smooth manifolds E,M, and F , and a smooth map π : E →→F M that ties them all together.

Definition 2.1.1 A smooth fiber bundle is a quadruple (π, E, M, F ) such that

1. F is a smooth manifold of dimension k called the model fiber;

2. M is a smooth manifold of dimension n called the base manifold;

3. E is a smooth manifold of dimension n + k called the total space;

4. π : E →→ M is a smooth surjective submersion such that for each point p ∈ M, the

−1 fiber Ep := π (p) is diffeomorphic to the model fiber F . The map π is called the bundle projection.

A fiber bundle will usually be denoted by π : E →→ M, sometimes shortening this to just π or E if we wish to stress the projection or total space, respectively. ∼ It is a fact that every bundle over a contractible base manifold is trivial: (E →→ M) = M ×F . Since, in general, the base manifold M is locally Euclidean, it follows that each point

5 in p ∈ M has a contractible neighborhood U ∋ p. Thus there is a neighborhood around each ∼ p ∈ M such that the restricted bundle E|U =: EU is trivial: EU = U × F . This property is called local triviality. It follows that M can be covered by an atlas of locally trivializing charts. This fact will be used extensively in the sequel.

In fact, this local triviality of fiber bundles is a special case of Ehresmann’s fibration theorem [17].

Theorem 2.1.2 Let M,N be manifolds, and suppose f : N → M is a proper surjective submersion; then f is a locally trivial fibration.

If π : E →→ M is a fiber bundle, then any smooth map s : M → E satisfying

π ◦ s = 1M is called a section of E (or of π). Notice that s is a right-inverse to the projection π. The space of all sections of E is denoted by ΓE. In general, ΓE is an infinite-dimensional manifold. This fact is worth noting, but will not be used in this dissertation. For more information, see [26].

Example 2.1.3 Let M be a smooth manifold. A smooth function on M is a section of the trivial line bundle M × R. The space of all smooth functions on M forms an algebra under pointwise addition, multiplication, and scalar multiplication. The function algebra on M is denoted by FM = C∞(M), or simply F if the manifold M is understood from context.

Example 2.1.4 Let π : TM →→ M be the tangent bundle of a manifold M. A section of

TM is called a vector field on M. The space of all vector fields is denoted by XM = Γ(TM), or simply X when M is fixed or clear from context.

The spaces of local smooth functions and vector fields over a trivializing chart Uα ⊂ M will be denoted by Fα = FUα and Xα = XUα, respectively.

Let M be a manifold, and (Uα, x) be a chart centered at p ∈ M. A map (x) =

1 n n (x , . . . , x ): Uα → R with x(p) = (0,..., 0) is called local coordinates on M about p. The

6 n restricted tangent bundle over Uα is trivial: TUα = Uα × R . The local coordinates on Uα

induce a basis for TUα, given by { } ∂ ∂ ,..., =: {∂ , . . . , ∂ } . ∂x1 ∂xn 1 n

Any tangent vector in TUα can then be written as a linear combination

1 2 n v = y ∂1 + y ∂2 + ··· + y ∂n.

We write v = (x, y) = (x1, . . . , xn, y1, . . . , yn). We shall call (x, y) the induced local compo-

1 n 1 n nents and (x , . . . , x , y , . . . , y ) the induced local coordinates of v ∈ TUα.

The second tangent bundle TTUα decomposes as follows:

∼ n TTUα = T (Uα × R )

∼ n = TUα × T R

∼ n n n = Uα × R × R × R .

1 n 1 n The local coordinates (x , . . . , x , y , . . . , y ) on TUα induce bases for the factors of TTUα

i e i given by ∂i := ∂/∂(x ◦ π) and ∂i := ∂/∂y , i = 1, . . . , n. Then any vector V ∈ TTUα can be written as a linear combination

1 n 1 e n e V = X ∂1 + ··· + X ∂n + Y ∂1 + ··· + Y ∂n. (2.1.1)

The local coordinate representation of V in this basis is given by

V = (x, y, X, Y ) = (x1, . . . , xn, y1, . . . , yn,X1,...,Xn,Y 1,...,Y n).

One refers to this representation as induced local coordinates for V over Uα. Einstein’s summation convention will be used when talking about vector fields, forms, and tensors. This convention states that whenever an index is repeated as a superscript and subscript in different factors, then one is to sum over that index. For example,

∑n i i 1 n a ∂i := a ∂i = a ∂1 + ··· + a ∂n. i=1

7 The vector of equation (2.1.1) is written in this convention as

i i e V = X ∂i + Y ∂i,

where this “+” is actually a direct sum, so it must be written explicitly.

Vectors of the form (x, y, 0,Y ) are called vertical, while those of the form (x, y, X, 0)

are called basal. Vertical vectors are natural in the sense that the form (x, y, 0,Y ) of their

induced local coordinate expression is preserved under change-of-charts on M. The same is

not true of basal vectors.

Example 2.1.5 Let π : E →→ M be a fiber bundle. Suppose (Uα, x) is a local coordinate

chart on M, and (Wβ, y) is a local coordinate chart on the model fiber F . Then (x, y) are

local coordinates on the slice Uα × Wβ of Eα = Uα × F , and they induce local coordinates | i ◦ e i on (TE) Uα×Wβ . The bases are given by ∂i = ∂/∂(x π) and ∂i = ∂/∂y , as above. Tangent vectors to E then have induced local coordinates of the form (x, y, X, Y ) over such a chart. ∼ If the model fiber F is parallelizable, i.e., if TF = F ×Rk, then these local coordinates

are defined at once on all of TEα.

2.2 Special Bundles

In this section we review a number of examples of bundles and related notions that

will be used throughout the paper. These include pullback bundles, lifts of paths, G-bundles,

associated bundles, and jets. A detailed treatment of all of these except jets may be found

in [26]. For jets, one is referred to [19] or [22].

Let f : N → M be a smooth map of smooth manifolds, and suppose that π : E →→ M is a fiber bundle over M. The pullback bundle of E along f is a fiber bundle f ∗π : f ∗E →→ N

given by

f ∗E := {(x, v) ∈ N × E | f(x) = π(v)},

and the bundle projection f ∗π : f ∗E →→ N is the projection onto the first factor of the

product N × E. The fibers of f ∗E over N are diffeomorphic copies of the fibers of E over

8 im f ⊆ M, ∗ ∼ (f E)p = Ef(p).

The following diagram commutes.

f f ∗E ♮ / E

f ∗π π   N / M f

∗ The map f♮ : f E → E is called the pushforth of f : N → M. The bundle morphism (f♮, f) is natural in the category of fiber bundles, hence the notation. More precisely, this means that any bundle morphism (u, f):(E1 →→ N) → (E2 →→ M) factors through the morphism

∗ (f♮, f):(f E2 →→ N) → (E2 →→ M).

∗ Let s ∈ Γf E be a smooth section over N. The pushforths ˜ = f♮s : N → E is said to be a section of E along f. The space of all sections along f : N → M is denoted by

∗ Γf E := f♮Γf E. Pullback bundles are especially useful in dealing with paths in M. Let γ : I → M be a smooth map with 0 ∈ I and γ(0) = p ∈ M. We say that im(γ) is a path in M emanating from p. We will frequently regard a given path emanating from p ∈ M as a map γ : I → M with I = [0, 1], or I = (−ε, 1 + ε) if necessary. In particular, we allow for paths in M to have self-intersections.

Let π : E →→ M be a fiber bundle, and γ : I → M a path in M with γ(0) = p and

γ(1) = q. Let c : I → γ∗E be a section of the pullback bundle γ∗π : γ∗E →→ I. The push forth γ♮c is a path in E that satisfies π ◦ γ♮c = γ, so that the following diagram commutes.

∗ γ♮ / γ E = E O {{ {{ γe {{ c {{ π {{ {{  { / I γ M

∗ The path γe = γ♮c : I → E is called a lift of γ to E. Every section c ∈ Γγ E corresponds to

∗ a different lift of γ. The space of all lifts of γ to E is denoted by ΓγE := γ♮Γγ E. As the

9 notation suggests, one can think of a lifted path as a section of E along γ. However, if γ has

self-intersections then this is not an honest section; hence the need for pullback bundles in

the first place.

Example 2.2.1 Let γ : I → M be a path in M, and consider the tangent bundle π : TM →→

M. Further, suppose im(γ) is contained in a trivializing chart (Uα, x) centered at γ(0) = p.

1 2 n For each t ∈ I, the local coordinates of γ(t) in Uα are given by γ(t) = (γ (t), γ (t), . . . , γ (t)).

∗ The velocity lift of γ to TM is given by γ♮c, where c : I → γ TM is given by ( ) dγ1 dγ2 dγn c(t) = (t), (t),..., (t) dt dt dt = (γ ˙ 1(t), γ˙ 2(t),..., γ˙ n(t)).

1 n 1 n We writeγ ˙ := γ♮c = (γ , . . . , γ , γ˙ ,..., γ˙ ) ∈ ΓγTM, orγ ˙ = (γ, γ˙ ) in component form.

The acceleration lift of a path γ : I → M is the velocity lift of its velocity lift. It is a

pathγ ¨ : I → TTM given in local coordinates by

γ¨ = (γ, γ,˙ γ,˙ γ¨).

We obtain the following tower of velocity lifts.

TTME

πT 

γ¨ TM< zz zz zz zz π zz γ˙ zz  z / I γ M

This tower may be extended vertically ad infinitum; however only the velocity liftγ ˙ itself has any geometric significance without adding structure to M. The acceleration liftγ ¨ will play a role in Chapter 5.

10 Let f : N → M be a smooth map of manifolds. Then there is an induced tangent

map T f = f∗ : TN → TM that makes the following diagram commute.

f∗ TN / TM

πN πM   N / M f

For any vector field X ∈ XN, the map f∗X : N → TM is analogous to the Jacobian of the map f applied to the vector field X. In induced local coordinates, it is given by

f∗X(p) = (f(p), (f∗X)p),

where the second component is given by the standard Jacobian

(f∗X)p = f∗p · Xp = Tpf · Xp = Df(p) · Xp.

Only the ∗-notation will be used to denote this map throughout the remainder of this dis-

sertation.

Of particular interest is the case when π : E →→ M is a fiber bundle, σ ∈ ΓE, and

X ∈ XM. Then σ∗ : TM → TE can be thought of as lifting the vector field X on M to a vector field on E, along σ. In induced local coordinates we have

σ∗X(p) = (p, σp,Xp, Dσ(p) · Xp),

and the following diagram commutes.

σ∗ / TMO TE

X πT  / M σ E

Let π : E →→ M be a fiber bundle with model fiber F , and let G be a Lie group.

Suppose that there is an effective action τ : F × G → F making the model fiber F a (right)

G-space. Then we may regard τ : G → Aut(F ) as an inclusion. If the bundle cocycle φαβ

11 takes values only in G, then we say that π : E →→ M is a G-bundle. The group G is called the structure group of the bundle.

Definition 2.2.2 If the model fiber of a G-bundle π : P →→ M is the group G itself, and the defining action of G on itself is the right regular representation, then (π, P, M, G) is called a principal G-bundle.

Now suppose that π : P →→ M is a principal G-bundle, and F is a left G-space. One may construct a new G-bundle over the base manifold M by collapsing the fibers of P at each point and replacing them with copies of F via the bundle’s G-cocycle. Indeed, define a right action of G on P × F by

(p, f)g := (pg, g−1f), and then define

P [F ] := (P × F )/G, the orbit space of this action. This makes π : P [F ] →→ M an associated G-bundle over M.

All G-bundles are obtained by this process: a G-bundle is either a principal G-bundle, or it is associated to a principal G-bundle. It follows that every principal G-bundle π : P →→ M is a functor from the category of (left) G-spaces to the category of (associated) G-bundles over M. We write P : F 7→ P [F ] to denote this relationship.

In fact, every bundle may be regarded as a G-bundle for some G. For any manifold

F , the space of automorphisms Aut(F ) forms a group, albeit an infinite-dimensional one in general. For any bundle with model fiber F , the cocycle φαβ always takes values in this group. Thus one may regard any bundle with model fiber F as an Aut(F )-bundle. However, the group Aut(F ) is in general too large for this fact to be useful. To extract any meaningful information about the bundle, it is necessary to reduce the structure group to some subgroup

G ≤ Aut(F ).

12 Example 2.2.3 Let M be an n-dimensional manifold. On an atlas {(Uα, φα)} of M, define the Leibniz cocycle by ( ) ◦ −1 ℓαβ(p) := D φα φβ (φβ(p))

n n for all p ∈ M, where D denotes the Jacobian. Since Dφαβ : R → R , we may identify

n each ℓαβ with an element of GL(R ) = GLn, making it a GLn-cocycle. The principal GLn- bundle with this Leibniz cocycle is called the bundle of linear frames over M. We write

π : LM →→ M to denote this bundle. The tangent bundle of M is the associated GLn-bundle

n n L[R ] =: TM, under the standard action of GLn on R .

In studying horizontal bundles and connections, the concept of 1-jets will be useful in extending local properties to global ones. This treatment follows [22, §12], where jets are studied in much more generality and detail.

Essentially, two functions f, g : M → N determine the same 1-jet at a point p ∈ M if f(p) = g(p) = q ∈ N, and Tq(im f) = Tq(im g) in TqN. Thus, the 1-jet of a function f : M → N at a point p ∈ M is analogous to the linearization of a map F : Rn → Rm at a point in Rn. This idea is made more precise.

Definition 2.2.4 Two curves α, γ : R → M are said to have first order contact (at zero) if for every smooth function φ ∈ F(M), the difference function δ := φ ◦ α − φ ◦ γ satisfies

′ d δ(0) = 0 and δ (0) = dt t=0(δ) = 0.

We write α ∼1 γ to denote this relationship. It follows that ∼1 is an equivalence relation on C∞(R,M). Indeed, reflexivity and symmetry are obvious. Transitivity follows from

(φ ◦ α − φ ◦ γ) − (φ ◦ δ − φ ◦ γ) = φ ◦ α − φ ◦ δ.

Lemma 2.2.5 If α ∼1 γ, then f ◦ α ∼1 f ◦ γ for every smooth map f : M → N.

Proof: If φ ∈ F(N), then φ ◦ f ∈ F(M). □

13 Definition 2.2.6 Two maps f, g : M → N are said to determine the same 1-jet at p ∈ M if for every curve γ : R → M with γ(0) = p, then f ◦ γ ∼1 g ◦ γ.

1 1 1 1 We write jp f = jp g, or j f(p) = j g(p), to denote this relationship. An equivalence class

1 under this relation is called a 1-jet of M into N. The 1-jet jp f depends only on the germ

[f]p of f at p. The set of all 1-jets of M into N is denoted by J 1(M,N). Given an element x =

1 ∈ 1 jp f J (M,N), the point p =: σ(x) is the source of x and the point f(p) =: τ(x) is the target of x. The maps σ and τ are called the source map and target map, respectively.

∈ 1 The set of all 1-jets from M into N with source p M is denoted by Jp (M,N), and ∈ 1 1 the set of all 1-jets with target q N is denoted by J (M,N)q. We write Jp (M,N)q := 1 ∩ 1 1 → 1 7→ 1 Jp (M,N) J (M,N)q. The map j f : M J (M,N): p j f(p) is called the first jet prolongation of f : M → N.

Let π : E →→ M be a smooth fiber bundle with model fiber F .A section of π is a

1 smooth map s : M → E such that π ◦s = 1M , so s is a right-inverse to π. We denote by J E (also written as J 1(π)), the set of all 1-jets of local sections of E. The set J 1E is called the

∈ 1 1 first jet prolongation of E (or of π). An element x Jp (M,E) belongs to Jp E if and only 1 ◦ 1 1 1 if jτ(x)π x = jp 1M . It follows that J E is a closed submanifold of J (M,E). Moreover, for every section s of π, j1s is a section of σ : J 1E →→ M.

Example 2.2.7 The first jet prolongation J 1E fibers over E via the target map projection

1 →→ 1 7→ 1 τ : J E E : jp s s(p), giving J E the structure of an affine bundle over E. By definition, the space J 1(M,E) coincides with the vector bundle T ∗M ⊗ TE = L(TM,TE).

→ 1 ◦ A 1-jet x : TpM TvE, where π(v) = p, belongs to J E if and only if π∗ x = 1TpM . The ∗ ⊗ kernel of such a projection induced by π∗ is Tp M TvF . Thus the preimage of 1TpM in ∗ ⊗ ∗ ⊗ Tp M TvE is an affine subspace modeled on the vector space Tp M TvF .

Since jets are defined at single points, then jet prolongation preserves composition of maps of manifolds. If f : N → M and g : M → K are smooth maps of manifolds, and if

14 f(p) = q ∈ M, then 1 ◦ 1 ◦ 1 jp (g f) = jq g jp f.

This implies that J 1 is a functor. To read more about exactly how this works, see [22].

15 CHAPTER 3

HORIZONTAL BUNDLES AND CONNECTIONS

In this chapter, the principal objects of study are defined, and we begin to study their geometry. The well known material in this chapter follows [25], and is adapted from [28].

3.1 The Vertical Bundle

Let π : E →→ M be a smooth fiber bundle with model fiber F . As the total space

E is itself a manifold, we may consider its tangent bundle πT : TE →→ E. Further, since

π : E →→ M is a smooth map of manifolds, the induced tangent map π∗ : TE → TM exists,

and the following diagram commutes.

TE π∗ / TM (3.1.1)

πT πM   / E π M

If one thinks of the fibers of E as being vertical over the horizontal base manifold M,

then there is a natural way to define vertical tangent vectors in TE.

Definition 3.1.1 The vertical bundle πV : V E →→ E is the kernel

V E := ker π∗ (3.1.2)

When the total space E is clear from context, the vertical bundle will be denoted simply by

V . The vertical bundle is the global subbundle of TE that consists of all vectors tangent to

the fibers of E. Naturality of V follows from the fact that it is defined as a kernel.

The vertical bundle V E over a fiber bundle π : E →→ M is natural with respect to

pull back.

16 Proposition 3.1.2 Let f : N → M be a smooth map of manifolds. Then V f ∗E and

∗ (f♮) V E are isomorphic and the following diagram commutes.

∼ f V f ∗E = / f ∗V E ♮♮ / V E HH ♮ HH HH HH ∗ H f♮ πV πV πV HH HH#   f f ∗E ♮ / E

f ∗π π   N / M f

Proof: The tangent bundle of f ∗E is given by

∗ T f E = {(u, v) ∈ TN × TE | f∗u = π∗v}.

∗ Thus if (u, v) ∈ T f E, then f♮∗(u, v) = u = 0 if and only if v ∈ V E. Therefore, the map

∗ f♮♮ :(u, v) 7→ v is an isomorphism of fibers V f E → V E along f♮. □

Suppose π : E →→ M is a vector bundle. In this case, the vertical space Vv at each ∼ point v ∈ E looks like a copy of the fiber Ep = F . Thus the composite vertical bundle

π ◦ πV : V E →→ M is isomorphic to the bundle E ⊕ E over M. Moreover, the bundle E ⊕ E

over M is equivalent to the pullback of E along its own projection π : E → M. This allows

one to define a bundle version of canonical parallel translation on a vector space.

Definition 3.1.3 Let π : E →→ M be a vector bundle. The fiber-isomorphism J : π∗E →

V E given by d J(u, v) = J v := (u + tv) , u dt t=0 and the fiber-isomorphism K : V E → E along π given by

K : Juv 7→ v

define a notion of parallel translation along the fibers of the total space E.

17 The map J translates a vector v ∈ Ep to the vector Juv ∈ Vu, thought of as being the same

vector v based at u ∈ Ep. The map K returns the vector Juv based at u ∈ Ep to the vector v based at 0 ∈ Ep. The following diagram summarizes this relationship [26]. ∗ π E WWW J lll  WWWWW ∼ lll  W=WWW vlll  WWWWW+  V E3 WWW  pr E ⊕ E 3 WWWW  J jj  BB 3 WWWWW jjjj  B 33  WWWW jjj  BB 3  W+ uj  Bpr2 3  1V E V YY  BB πV 3  E4 YYYYY  B 3  4 YYYYY K BB 3 Ø 44  YYYYYY B (3.1.3) 4  YYYYYY! WW 44  , E WWW ππV 4  pr E WWWWW 4  WWWW 4 × π WWWWW+ WWW π M WWWW WWWW WWWW  1M WW+ M These maps are used extensively in studying the geometry of horizontal bundles and con-

nections on vector bundles. Unfortunately there are no maps akin to J and K for general

fiber bundles.

3.2 Horizontal Bundles

While the vertical bundle is naturally well defined for any smooth fiber bundle, there

is in general no natural way to define horizontal tangent vectors in TE. A horizontal bundle

on E is a choice of horizontal tangent planes in each fiber TvE.

Definition 3.2.1 A horizontal bundle on a smooth fiber bundle π : E →→ M is a subbundle

H E of TE that is complementary to the vertical bundle, so that

TE = H E ⊕ V E = H ⊕ V . (3.2.1)

In the extant literature, a horizontal bundle H on E is also commonly referred to as a

general connection on E. I have chosen to stress the subbundle of TE in this dissertation,

reserving the term connection in case H has more structure than simply being a splitting

of the tangent bundle.

Any n-plane subbundle of TE is known as an n-distribution on E. Thus a horizontal

bundle on E is also referred to as a horizontal distribution on E by some authors.

18 As V = ker π∗, it follows that for every v ∈ E, π∗ : Hv → TpM is a vector space isomorphism. Thus each subspace Hv of TvE looks like a copy of the tangent space TpM, possibly twisted with respect to local trivializations of E at p ∈ M. This relationship to the base manifold justifies calling H a horizontal subbundle of TE. We call the fiber Hv the horizontal space to E at v.

Theorem 3.2.2 Every smooth fiber bundle π : E →→ M admits a horizontal bundle H complementary to V .

Proof: Since the total space E is a manifold, it admits an auxiliary Riemannian metric V | g. Each fiber Ep is an embedded submanifold whose tangent bundle is T (Ep) = Ep . The ≤ | V | metric g determines a normal bundle N(Ep) TE Ep that is complementary to Ep by

definition. Define the horizontal spaces at each point v ∈ Ep to be Hv := Nv(Ep). These spaces vary smoothly along fibers of E by definition, and between fibers of E since g is

smooth. □

A typical fiber of TE looks like a copy of Rn+k. Because V is natural, the vertical

space Vv is a fixed k-plane in the fiber TvE. A horizontal space Hv is a choice of an n-plane

such that Hv +Vv = TvE. To make a horizontal bundle over E, these horizontal spaces must vary smoothly as v varies in E. This situation can be described using a special Grassmann

bundle over E.

n+k Recall that the Grassmannian G(n, R ) =: Gn(n+k) is the space of all n-dimensional

n+k ∼ n+k subspaces (or n-planes) of R . In a single fiber TvE = R , the vertical space Vv is a

fixed k-plane. A given n-plane H ∈ Gn(n + k) is admissible as a horizontal space in TvE if

and only if it is complementary to Vv. We define the horizontal Grassmannian at v ∈ E to

be the subspace of Gn(n + k) consisting of all admissible n-planes.

Definition 3.2.3 The set of all n-planes in Gn(TvE) that are complementary to Vv is called

the horizontal Grassmannian at v ∈ E, and is denoted by GH (TvE) ≤ Gn(TvE).

19 The horizontal Grassmann GH (TvE) is an open submanifold in Gn(TvE) since Vv is natural

(fixed) in TvE.

n+k Let LE denote the linear frame bundle over E. The defining action of GLn+k on R

n+k makes LE a principal GLn+k-bundle over E, whence TE = L[R ] is the associated bundle, as in Example 2.2.3. This action also induces the standard action of GLn+k on Gm(n + k) for any m ∈ [0, n + k], allowing one to define Grassmann bundles over E.

Definition 3.2.4 The associated bundle L[Gm(n+k)] =: Gm(TE) is called the Grassmann m-plane bundle over E, and GH (TE) := L[GH (n + k)] is called the horizontal Grassmann bundle over E.

Proposition 3.2.5 A horizontal bundle H is completely determined by a section of the horizontal Grassmann bundle GH (TE).

Proof: A section Γ ∈ ΓGH (TE) picks out a horizontal space over every point v ∈ E that is complementary to Vv. The horizonal spaces vary smoothly as v varies in E since Γ is smooth, whence the horizontal bundle is given by H = im(Γ). □

A horizontal bundle can also be determined by specifying the projection from TE =

H ⊕ V to either of its factors.

Proposition 3.2.6 A horizontal bundle H is completely determined by specifying the hor- izontal projection H : TE →→ H . The projection H must satisfy the following axioms, whence the horizontal bundle is given by H := im H.

H1. H ∈ Γ End(TE), so H is a tensor on E;

H2. H2 = H;

H3. ker H = V .

20 Proof: The first two axioms guarantee that H is a projection on each fiber of TE, and

varies smoothly from fiber to fiber. Therefore H = im H is a smooth subbundle of TE.

The third axiom ensures that H is complementary to the vertical bundle V , hence H is a horizontal bundle on E. □

Proposition 3.2.7 A horizontal bundle H is also completely determined by specifying the vertical projection V : TE →→ V . In this case, the horizontal bundle is given by H = ker V.

Axioms for V are:

V1. V ∈ Γ End(TE), so V is a tensor on E;

V2. V2 = V;

V3. im V = V .

Proof: Repeat the proof of Proposition 3.2.6, mutatis mutandis. □

3.3 Connections and Parallel Transport

This section describes the exact manner in which a suitable horizontal bundle deter- mines a system of parallel transport in the total space of a bundle, along the base manifold.

Let π : E →→ M be a smooth fiber bundle with model fiber F , H a horizontal bundle on E,

and V the vertical projection of Definition 3.2.7 that is associated to H .

Let γ : I → M be a path in M with γ(0) = p and γ(1) = q, and consider the pullback

bundle γ∗π : γ∗E →→ I. Recall that a lift of γ to E is a path γe : I → E that makes the following diagram commute.

∗ γ♮ / γ E {= E { γe { ∗ { γ π { π  {  { / I γ M In particular, the domain of a lifted path γe must coincide with the domain of the path γ in the base manifold.

21 ˙ Definition 3.3.1 A lift γe : I → E in Γγ E is said to be horizontal if and only if γe(t) ∈ Hγe(t) for all t ∈ I. A horizontal lift of γ is denoted by γe = γ.

˙ Alternatively, a lift γe ∈ Γγ E is horizontal if and only if its velocity lift γe is a path in H , γe˙ : I → H .

Definition 3.3.2 A path γ : I → M is said to have horizontal lifts if and only if for every

v ∈ Ep there is a unique horizontal lift γ : I → E such that γ(0) = v and γ(1) ∈ Eq.

In other words, a path γ has horizontal lifts if for every initial value v ∈ Ep, there is a unique horizontal lift emanating from v that extends over the entire domain of γ.

∗ By Proposition 3.1.2, pullback is a functor and V is natural, so that (γ♮) V E = V (γ∗E). Therefore the following diagram commutes.

(γ )∗ V (γ∗E) ♮ / V E

∗ (γ♮) πV πV   γ γ∗E ♮ / E

γ∗π π   / I γ M

∗ ∗ ∗ ∼ k+1 Now Vv(γ E) = (γ♮) (VvE) is a fixed k-plane in Tv(γ E) = R . The vertical projection V : TE →→ V E pulls back to a vertical projection on T (γ∗E), given by

e ∗ ∗ ∗ V : T (γ E) → V (γ E): x 7→ (γ♮) V(γ♮∗x). (3.3.1)

The kernel of Ve is a horizontal bundle Hf := ker Ve on γ∗E induced by H . In ( ) f particular, if im(γ) has no self-intersections, then H is equivalent to H ∩ T E|im(γ) , the restricted bundle over γ. Using a pullback makes this induced horizontal bundle well defined

for paths that may have self-intersections.

22 The induced horizontal bundle Hf is a line bundle over γ∗E = I × F . Since all line bundles are integrable, Hf has integral curves in γ∗E. Moreover, the integral curves of Hf intersect the fibers of γ∗E transversally since T (γ∗E) = Hf⊕ V (γ∗E). Thus, the integral curves of Hfdetermine a horizontal foliation of γ∗E whose leaves are transverse to the fibers.

∗ Let Pγ denote the horizontal foliation of γ E, and Pγ v the unique leaf that passes through v ∈ Ep. Definition 3.3.2 can be restated in terms of this horizontal foliation.

∗ Proposition 3.3.3 A path γ : I → M has horizontal lifts if and only if γ π(Pγ v) = I for all v ∈ Ep. □

In 1950, Ehresmann [17] included in his definition of a connection this condition that the leaves of Pγ project to all of I for all paths γ : I → M.

Definition 3.3.4 A horizontal bundle H on E is said to have horizontal path lifting (HPL) if and only if every path γ : I → M has horizontal lifts. A horizontal bundle with HPL is called an (Ehresmann) connection on E.

Some sources including [22, 29] call a horizontal bundle with HPL a complete connection.

We shall reserve this term for another, a priori unrelated, notion.

Horizontal bundles with HPL are called connections because the horizontal lifts of a path γ in M can be used to connect the fibers of the total space E along γ. This works as follows.

∗ If γ has horizontal lifts, then Pγ v can be thought of as the image of a section in Γγ E satisfying Pγ v(0) = v ∈ Ep. The unique horizontal lift of γ with initial value v ∈ Ep is then given by the pushforth γ = γ♮(Pγ v): I → E.

∗ γ♮ / γ E = E O {{ {{ {{ P { γ v {{ π {{ γ {{  { / I γ M

23 In this case, Pγ determines a diffeomorphism between the fibers Ep and Eq. With just a little abuse of notation, we write

Pγ : Ep → Eq : v 7→ (Pγ v)(1), (3.3.2)

and call Pγ parallel transport along γ.

Definition 3.3.5 If H is an Ehresmann connection, then there is parallel transport Pγ along every path γ : I → M. The collection of all such diffeomorpshisms P = {Pγ | γ : I → M} is called (a system of ) parallel transport in E, along M.

Proposition 3.3.6 This parallel transport P in E, along M, has the following properties.

1. Existence and uniqueness: for each v ∈ Ep and each smooth γ with γ(0) = p, there

exists a unique smooth horizontal curve Pγ v from v ∈ Ep to Eq. Alternatively, one

may regard Pγ v as a horizontal section of E along γ ( via pullback if necessary).

2. Invertibility: allowing v to vary over Ep, the resulting map Pγ : Ep → Eq is a diffeo- morphism. Its inverse is parallel transport along the reverse ←γ(t) := γ(1 − t).

3. Parametrization independence: if δ is a reparametrization of γ, then Pδ = Pγ : Ep →

Eq.

4. Smooth dependence on initial conditions: For every trivializing open set Uα in M and

each smooth map f : TUα → M with f(p, 0) = p for all p ∈ Uα, the bundle map

× → 7→ P F : TUα Uα Eα Eα :(x, v) γ v(1),

where γ(t) := f(tx), is smooth.

5. Initial uniqueness: if γ and δ are two curves emanating from p ∈ M with γ˙ (0) = δ˙(0),

then for every v ∈ Ep, Pγ v and Pδ v have the same initial tangent vector; that is, ˙ ˙ Pγ v(0) = Pδ v(0).

24 Proof: Properties 1, 2, and 3 follow directly from the definition of the foliation Pγ . To see that properties 4 holds, suppose γ(t) = f(tx) where f(p, 0) = p. The initial condition

· −1 f ∗ for P v is (P v) = π∗| (x) ∈ H for each v ∈ γ E . This will vary smoothly as H is γ γ Hv v p

smooth. Similarly, suppose γ(t) = f(tx) and δ(t) = g(tx) for f, g : TUα → M satisfying

· f(p, 0) = g(p, 0) = p. The curves Pγ v and Pδ v have the same initial condition, (Pγ v) =

· −1 (P v) = π∗| (x), and thus initially coincide. δ Hv □

The properties of this proposition may be taken as axioms for a system of paral-

lel transport in E, along M. Any family of fiber-diffeomorphisms satisfying these axioms

determines a connection on the bundle π : E →→ M.

∼ Proposition 3.3.7 Suppose P = {Pγ : Eγ(0) = Eγ(1) | γ : I → M} is a system of fiber- diffeomorphisms in E satisfying the implications of Proposition 3.3.6. Then P induces an

Ehresmann connection H on E.

∗ Proof: Properties 1, 2, and 3 imply that P determines a transverse foliation Pγ of γ E

∗ whose leaves Pγ v intersect every fiber of γ E, for every path γ : I → M. These foliations then determine the horizontal spaces of a horizontal bundle along γ by Hf = T (P), and

H ∩ T (E|im(γ)) = γ♮∗T (P). Property 5 implies that these foliations are compatible in the following sense. Let

γ, δ be two paths in M emanating from the same point, with the same initial tangent vector,

f −1 −1 γ˙ (0) = δ˙(0) = v; then H = π∗| (γ ˙ (0)) = π∗| (δ˙(0)). v Hv Hv Property 4 implies that the horizontal spaces determined by these foliations vary smoothly as im(γ) varies smoothly in M, and v varies smoothly in E. The horizontal spaces

Hv are then determined by { } f Hv = γ♮∗Hv | γ : I → M, ∥γ˙ (0)∥ = 1, v ∈ Eγ(0) . (3.3.3)

The horizontal lifts along each path γ : I → M are given by the foliation Pγ . □

25 f f +∞ ...... +∞ ¡ ¡ ¡ ¡¡ ...... ¡ ¡ ¡ ...... ¡ ¡ ¡ ¡ ...... ¡ ¡ ¡ ...... ¡ ...... ¡ ¡ ¡ ¡ ...... ¡ ¡ ¡ ...... ¡ fiber ¡ ¡ ...... ¡ ¡ fiber ...... ¡ ¡ over p ¡ ¡ ...... over q ¡ ¡ ...... ¡ ¡ ¡ ...... ¡ ¡ ...... ¡ ¡ ...... ¡ ¡ ¡ ¡ ...... ¡ ¡ ¡ ...... ¡ ¡ ¡ ...... ¡ ...... ¡ ¡ ¡ ¡ −∞ f...... ¡ ¡ ¡ ¡ f −∞

Figure 3.1: The horizontal foliation of a horizontal bundle on T R that is not a connection.

Corollary 3.3.8 Let P be a system of fiber-diffeomorphisms satisfying the implications of

Proposition 3.3.6, and H the connection of (3.3.3) induced by P of equation (3.3.3). The parallel transport determined by H is exactly P.

Proof: This follows immediately from the construction of H . □

We have thus shown that a horizontal bundle on E with HPL is equivalent to a system

of parallel transport in E, along M.

Theorem 3.3.9 An Ehresmann connection on E is a system of parallel transport in E,

along M. □

That a horizontal bundle has HPL is a nontrivial property. To see this we need only

consider the simplest kind of horizontal bundle: an integrable H on E.

Definition 3.3.10 A horizontal bundle H on a fiber bundle π : E →→ M is said to be flat

if and only if it is integrable.

One might suspect that every flat horizontal bundle has HPL, but this is not the case. The

integral manifolds of H determine a horizontal foliation of E with each leaf intersecting the

fibers of E transversally. However, it is possible that (at least one) leaf of this horizontal

foliation does not intersect every fiber of E.

26 Example 3.3.11 Consider π : T R →→ R as the trivial line bundle over R. Let γ : I → R

be the identity path γ(t) = t, and consider the horizontal foliation Pγ depicted in figure 3.1.

The horizontal bundle corresponding to this foliation is given by H = T (Pγ ). The solid lines begin in one fiber and go away to infinity. The dotted lines do not

intersect either fiber. Notice that there exists a unique horizontal lift of γ for every initial

value in TpR, but no horizontal lift of γ reaches from over p to over q.

The question of which horizontal bundles actually do have horizontal path lifting,

and hence are connections, is answered in section 6.2.

3.4 Kinds of Connections

If the fibers of E have an algebraic structure on them, then one can define different types of Ehresmann connections on E that respect this structure. These are the most

common connections found in the literature. This follows [13, 25].

Example 3.4.1 Let F be a G-space, making E a G-bundle over M. A horizontal bundle

H on E is said to be an (Ehresmann) G-connection if and only if H is G-equivariant.

This G-equivariance of H can be understood in terms of the vertical projection

V : TE →→ V . Let τ : G × F → F be an action of G on F . It is customary to identify the

group element g ∈ G with the diffeomorphism τg ∈ Aut(F ). A horizontal bundle H on a

G-bundle E is G-equivariant if and only if Vgv = g∗Vv. The horizontal spaces and horizontal

projection also satisfy Hgv = g∗Hv and Hgv = g∗Hv, respectively.

Example 3.4.2 If a horizontal bundle H is a G-connection on a principal G-bundle π :

P →→ M, then H is called a principal connection on (π, P, M, G). Principal connections

play an important role in high energy particle physics and gauge theories.

Example 3.4.3 Let V be an n-dimensional real vector space with general linear group

GL(V ). If π : E →→ M is a vector bundle with model fiber V , then a GL(V )-connection on

E is called a linear connection on E. These are the most studied connections, as the tangent

27 bundle TM of a manifold M is a vector bundle with model fiber Rn. Since every Levi-

Civita connection on a (pseudo-) Riemannian manifold is linear, therefore all Levi-Civita

connections are GLn-connections on TM. All other horizontal bundles on a vector bundle are called nonlinear. This includes the affine connections: G-connections on E with G = An, the affine group. It is unfortunate that linear connections are sometimes called affine connections in the literature [21, 2, 31, 32, 15], and a nonlinear connection is defined to be a highly restricted type of connection on TM ¬ 0.

Thus, one must read the context carefully to know which version of “affine” or “nonlinear”

connection a particular author is using.

Example 3.4.4 A Koszul connection is a linear operator of the type of a covariant deriva-

tive on a vector bundle. It gives rise to a linear connection on the vector bundle [28].

Example 3.4.5 A Cartan connection is similar to a principal connection, but the geometry

of the principal bundle is tied to the geometry of the base manifold [7, 30]. In particular,

Cartan connections describe the geometry of manifolds modeled on homogeneous spaces.

Intuitively, they model one manifold “rolling over” another one. Under certain technical

conditions, they can be related to the remaining types [30]. Cartan connections are not

studied in any detail in this dissertation.

28 CHAPTER 4

LOCAL DESCRIPTION: WONG ANGLES AND CHRISTOFFEL FORMS

In this chapter, we take a closer look at the relationship between a horizontal bundle

H on E and the natural vertical bundle V E. This is done by looking locally, over trivializing

charts Uα in M. 4.1 Local Subbundles of TE

Recall that a horizontal bundle H is a global subbundle of TE that is complementary

to the vertical bundle V . While V is naturally well defined for every smooth fiber bundle,

there is in general no natural way to define H . ∼ ∼ The exception is when E is a trivial bundle, E = M × F . In this case TE = TM × TF = TM ⊕ V , and the tangent bundle TM is itself a horizontal bundle on E called the canonical flat connection. Since this canonical connection is the tangent bundle to the base manifold M, we write TE = B ⊕ V and call B = TM the basal subbundle of TE.

Most fiber bundles are not trivial, but all fiber bundles are locally trivial. Thus one can describe a horizontal bundle H locally over a trivializing chart in M by comparing it

to the canonical connection over that chart.

{ } | ∼ × Let B = (Uα, φα) be a bundle atlas for E, and let Eα := E Uα = Uα F be the

local trivialization of E over Uα. The tangent bundle to Eα is

∼ T (Eα) = T (Uα × F ) ∼ = T (Uα) × TF (4.1.1)

=: Bα ⊕ Vα.

Since V is natural, it is not necessary to write the subscript α. However, we shall do so to

emphasize that this decomposition is purely local.

29 4.2 Wong Angles in TvE

To compare the vertical space Vv and horizontal space Hv in a single fiber TvE, it will be necessary to define a notion of “distance” between them. This is done using a collection of angles adapted from those used by Wong to study the geometry of Grassmann manifolds in

[34]. The idea to use these angles in the present context is originally due to Parker [25, 27].

Let π : E →→ M be a smooth fiber bundle over an n-dimensional base manifold M, with k-dimensional model fiber F . For any v ∈ E, the tangent space TvE looks like a copy

n+k of R . The vertical space Vv is a fixed k-plane, and a horizontal bundle H determines a complementary n-plane Hv in TvE.

n+k Suppose we endow TvE = R with a positive definite inner product gv = ⟨ , ⟩. For any two vectors u, w ∈ TvE, define the angle θ between u and w with respect to gv via the equation |⟨u, w⟩| cos θ = , (4.2.1) ∥u∥ ∥w∥ where ∥u∥ = ⟨u, u⟩1/2, so that θ ∈ [0, π/2]. This agrees with the standard notion of an angle between two lines in Euclidean space.

⊥ Now fix u ∈ Hv and let u ∈ Vv be its orthogonal projection onto Vv. The angle between u and u⊥ defined by equation (4.2.1) is a smooth function of u, ( ⟨ ⟩ ) u, u⊥ θ(u) = cos−1 . (4.2.2) ∥u∥ ∥u⊥∥

Allowing u to vary in Hv, the nonzero critical, or stationary, values of θ are called the principal Wong angles between the horizontal space Hv and vertical space Vv in TvE.

Since Hv and Vv are complementary vector subspaces of TvE, it follows that there are exactly m = min{n, k} principal Wong angles between Hv and Vv. For our purposes it will be useful to always have a collection of n = dim M Wong angles at every point in E.

1 m H V Definition 4.2.1 Let θv, . . . , θv denote the m principal Wong angles between v and v

k+1 n as measured by gv. If m = k, put θv , . . . , θv = π/2. We denote the family of n Wong

30 angles at v ∈ E by { } 1 2 n Θv := θv, θv, . . . , θv . (4.2.3)

Moreover, we call θv := min Θv the (smallest) Wong angle for H at v ∈ E.

Clearly, the values of the Wong angles depend explicitly on the choice of inner product gv. However, the stationary points of (4.2.2) are the same for any choice of inner product, modulo scalar multiples. If n ≤ k, these stationary vectors {u1, u2, . . . , un} determine a principal basis for the horizontal space Hv. If n > k, then the collection {u1, u2, . . . , uk} may be extended to a basis for Hv.

4.3 Wong Angles Along Ep

We shall need to compare Wong angles at different points along fibers of E. Let v ∈ E, and Uα be a trivializing chart centered at p = π(v) ∈ M. Let gF be a complete

Riemannian metric on the model fiber F , and gα the standard Euclidean metric on Uα. Now consider the product metric g := gα ×gF on Eα = Uα ×F . This makes (Eα, g) a Riemannian manifold.

To compare Wong angles in different fibers of TE along Ep, consider the Levi-Civita connection ∇g for the metric g. It is well known that ∇g determines a system of parallel

g transport P on Eα; in particular along the fiber Ep.

Fix v ∈ Ep and let δ : I → Ep be a path with δ(0) = w and δ(1) = v. Denote this path by δ : w 7→ v. Let θv(w, δ) denote the smallest Wong angle between the n-plane P g H P g V ∈ δ ( w) and k-plane δ ( w) in the fiber TvE as measured by gv. For every w Ep, the

Wong angle θv(w, δ) > 0 since Hw and Vw are complementary in TwE. However, as w varies along Ep, it is possible that Hw could become asymptotic to Vw. Along a single fiber Ep, the Wong angles θv(w, δ) are bounded below by { }

θp = θ(Ep) := inf inf θv(w, δ) ≥ 0. (4.3.1) w∈Ep δ:w7→v

Notice that the numerical value of θp depends implicitly on the metric g. However, whether

θp is bounded away from 0 is independent of the choice of metric.

31 Proposition 4.3.1 If θp is bounded away from 0 with respect to one metric of the form g = gα × gF , then it is bounded away from zero with respect to every local trivialization

Uα × F containing Ep, and every choice of complete Riemannian metric gF on F .

Proof: Regardless of the choice of chart Uα, the product metric g = gα × gF always makes

Bα orthogonal to Vα. Thus, we need only show that θp being bounded away from 0 is independent of choice of fiber-metric gF .

Let u1, . . . , um be the principal directions determined by principal the Wong angles

1 m θ , . . . , θ . That is, u1, . . . , um are the nonzero stationary points of the function θ defined ∥ ∥ in equation (4.2.2). Suppose further that ui g = 1 for all i = 1, . . . , m. Let b1, . . . , bm be the projections of the ui onto the basal space Bα. Each bi is a nonzero vector since H is complementary to V . The lower bound θp will be bounded away from 0 if and only if

∥bi∥ is bounded away from 0 along the fiber Ep. Clearly this is independent of the choice of

fiber-metric gF . □

Notice that θ = θv can be thought of as a smooth function in F(Ep) that takes values in [0, π/2]. It is bounded below by θp.

Proposition 4.3.2 Let gF be a complete Riemannian metric on the model fiber F of a fiber bundle π : E →→ M. Then the local metrics g = gα × gF on Eα can be combined to form an honest Riemannian metric β on all of E.

This follows from a standard partition of unity argument.

Proof: Let {Uα, φα} be an atlas on M and suppose that {ψα} is a partition of unity on

M subordinate to the cover {Uα} by refining {Uα} if necessary. On each local trivialization

Eα = Uα × F define a product metric gα × gF , as above. Then the sum ∑ ∑ β = ψα(gα × gF ) = ψαgα × ψαgF α α is a metric on E. □

32 Along any fiber Ep, the metric β is a finite sum of product metrics with vertical factor ∑ ψαgF = gF . Therefore the Wong angles along every fiber of E can be measured in this metric β. Thus the local basal subbundle Bα over any trivializing chart Uα may be replaced by the normal bundle Bp := N(Ep) determined by β when only the fiber Ep is of interest.

4.4 Local Christoffel Forms

In general, a horizontal bundle Hα has components in both Bα and Vα over Eα. Let

(p, v) ∈ Eα, and suppose {X1,...,Xn} is a basis for Bv = Tp(Uα) and {V1,...,Vk} is a basis for Vv = TvF . Then any vector Hv ∈ Hv can be written as a linear combination of vectors in Bα and Vα,

i j Hv = a Xi + b Vj.

i In particular, suppose X = a Xi is a fixed vector in Bv. Then there is a unique horizontal vector Hv ∈ Hv with the same basal component as X. Its vertical component depends on both X and the point v ∈ Ep. We write

α Hv = X + Γ (v, X),

α where Γ (v, X) = Vv is the unique vertical component. Consider v ∈ F as a constant section in ΓEα. Then, for any X ∈ Xα, v∗X has local coordinates (p, v, Xp, 0) in TvE, and the map Γα is given by α α −V Γv X := Γ (v, X) = (v∗X).

α Definition 4.4.1 Let Uα be a trivializing chart in M for E. The map Γ :ΓEα ×XUα → Vα given in local coordinates over Uα by

α −V Γ (σ, X)(p) = σp (p, σp,Xp, 0)

is called the local Christoffel form for H on Uα.

Fixing v ∈ F , the local Christoffel form

α α → 7→ −V Γv := Γ (v, ): Xα XF : Xp v(p, v, Xp, 0)

33 1. Vv 2. Vv Hv Hv

Bα Bα

Xp Xp

Vv(Xp, 0)

3. Vv 4. Vv Hv Hv α Γv Xp

Bα Bα Xp Xp

α Figure 4.1: The action of a Christoffel form Γv on a single vector Xp.

is a V -valued 1-form on Uα, justifying the name. This form measures the failure of H to

α be trivial along vector fields on Uα. Figure 4.1 depicts the action of a Christoffel form Γ in the slice of TvE determined by the vector Xp.

α The Christoffel form Γ completely determines the connection H over Uα. With only a little abuse of notation,

α Hv = Bα + Γ (v, Bα), or allowing v to vary in F , H | B α B Eα = α + Γ (F, α).

The local Christoffel forms over two different charts in M are related via the bundle cocycle. Let Uα and Uβ be two trivializing charts at p ∈ M with transition map φαβ ∈

Aut(Ep), and suppose X ∈ X(Uαβ). The local Christoffel forms satisfy

β α (φαβ)∗ ◦ Γ = Γ ◦ φαβ.

34 β α That is to say that Γ and Γ are φαβ-related, and the following diagram commutes.

V (φαβ )∗ / V Oβ Oα

Γβ Γα

X(U ) × E / X(U ) × E β β φαβ α α

The local Christoffel forms are tensorial in v since V is, but they are merely smooth in

X. Therefore (φαβ)∗ must obey the chain rule in X, yielding the following change-of-charts formula.

Proposition 4.4.2 Let Uα and Uβ be trivializing charts for E at p ∈ M with transition map

α β φαβ. The local Christoffel forms Γ and Γ satisfy

α β Γ (φαβ(p, v),Xp) = φαβ(p, )∗v ◦ Γ (v, Xp) + φαβ( , v)∗p ◦ Xp.

Proof: For (X,Y ) ∈ TvE, the tangent map of the change of charts map φαβ acts like

φαβ(p, v)∗(X,Y ) = (X, φαβ(p, v)∗(X,Y )).

This follows from the fact that φαβ(p, v) = (p, φαβ(p, v)). Now the Christoffel forms satisfy

−1 β −V −1 (φβ )∗(0, Γ (v, X)) = ((φβ )∗(X, 0)) ( ) −V −1 ◦ −1 = (φα )∗(φα φβ )∗(X, 0) ( ) −V −1 = (φα )∗(X, (φαβ)∗(X, 0) ( ) ( ) −V −1 − V −1 = (φα )∗(X, 0) (φα )∗(0, (φαβ)∗(X, 0) −1 α − −1 = (φα )∗ (0, Γ (φαβ(p, v),X)) (φα )∗(0, (φαβ)∗(X, 0)).

Applying (φα)∗ to both sides of the equation, looking only at the vertical component, and

α rearranging to solve for Γ (φαβ(p, v),X) yields the change-of-charts formula of the proposi- tion. □

∈ α For each v Eα, the Christoffel form Γv is a vertical-valued 1-form on Uα, but

α ∗ allowing v to vary in Eα makes Γ a section of the bundle T Uα ⊗ Vα →→ Eα. By Example

35 1 2.2.7, this bundle is equivalent to the first jet bundle of Eα over Uα, τ : J (Eα) →→ Eα. The 1 ∈ projection τ is the target map, τ(jp s) = s(p) Ep. Thus, one may consider a horizontal bundle locally as a section of the first jet prolongation of E over Uα. The advantage to thinking in terms of jets is that J 1 is a functor and the sections Γα are natural [22]. This means that, while each Christoffel form is a local section of J 1E, all of the data encoded by the collection of Christoffel forms on M can be collected in a single global section of J 1E. The following diagram illustrates the situation.

1 1 J (ι) / 1 J (EO α) J OE

Γα Γ

/ Eα ι E

Since the Christoffel forms locally determine the horizontal bundle H , this gives an alternate characterization of a horizontal bundle.

Proposition 4.4.3 A horizontal bundle H = Γ on a fiber bundle π : E →→ M is a section of the first jet prolongation of E, Γ ∈ Γ(J 1E →→ E). □

The restriction of Γ to Eα over a trivializing chart Uα yields the Christoffel form α | Γ = Γ Eα . This can be easily visualized by replacing the inclusion ι by the projection prα and turning around the arrows in the commutative diagram above.

Local Christoffel forms and Wong angles are two different means of studying a general connection along a fiber of E. As one may guess, these two notions can be related. For this purpose, it will be useful to use the n angles between the n-planes Hv and Bp. These are defined to be the complements of the Wong angles Θv.

Definition 4.4.4 The complementary Wong angles between Hv and Bp are given by

c i − i (θv ) := π/2 θv,

c c − i = 1, . . . , n. The collection of all n is denoted Θv , and the largest is θv = π/2 θv.

36 Vv

θv Hv

c θv Bp

Figure 4.2: Wong angles in a 2-dimensional slice of single fiber TvE.

c c − Clearly, the complementary angles θv are bounded above by θp = π/2 θp along a fiber

Ep. Defining the complementary Wong angles this way is natural, given the definition of

the Wong angles. Since Bp is orthogonal to Vv with respect to gv, the algebraic direct

sum decomposition TvE = Bp ⊕ Vv is actually an orthogonal sum. Figure 4.2 depicts the relationship between the Wong angle and its complement in a 2-dimensional slice of a single

fiber of TE.

∈ α α Recall that for fixed v Eα, the Christoffel form Γ (v, ) = Γv is a smooth map

α ∼ Rn → V ∼ Rk 7→ α Γv : TpUα = v = : Xp Γv Xp = Vv.

α × The norm of Γv , as measured with respect to g = gα gF , is given by

α α ∥Γ ∥ = sup ∥Γ Xp∥ . (4.4.1) v gv v gF ∥X ∥ =1 p gα

α If we allow v to vary over a single fiber Ep, then the norm of Γ along Ep is

∥Γα∥ := sup ∥Γα∥ . (4.4.2) Ep v gv v∈Ep

The norm of Γα along a fiber of E can be formulated in terms of the (complementary)

Wong angles along that fiber. Let Xp be a unit vector in Bp, v ∈ Ep, and Hv := Xp + α ∈ H α Γv (Xp) v. If θ is the angle between the vectors Hv and Xp, then the norm of Γv (Xp) is

37 given by ∥ α ∥ Γv Xp = tan(θ). (4.4.3)

c Since such angles are bounded above by θv , we obtain

∥ α∥ c Γv = tan(θv ). (4.4.4)

c ≥ c ∈ c Now, allow v to vary along Ep and recall that θp θv for all v Ep. If θp < π/2, we obtain

α a bound for the norm of Γ along Ep,

∥Γα∥ = tan(θ c). (4.4.5) Ep p

c α If θp = π/2, then Γ is unbounded along Ep. We have proved the following proposition.

Proposition 4.4.5 Let Eα be a local trivialization of E around p ∈ M, and gF any complete

α Riemannian metric on F . The local Christoffel forms Γ are bounded along Ep with respect to gF if and only if the Wong angles are uniformly bounded away from 0 along Ep, θp ≥ ε > 0.

□

4.5 Horizontal Spaces Along a Lifted Path

Lifts of paths from M to E play a prominent role in differential geometry. Because of their role in defining parallel transport, horizontal lifts (Definition 3.3.1) are especially important. In this subsection, we determine exact criteria for an arbitrary lifted path to be horizontal using the local Christoffel forms.

Suppose Uα ⊆ M is a trivializing chart at p for both E and TM, and Wβ ⊆ F is a | trivializing chart at v for TF . Then T (E Uα×Wβ ) := T (Eα,β) can be written as

T (Eα,β) = T (Uα × Wβ)

= T (Uα) × T (Wβ)

= Bα ⊕ Vβ (4.5.1)

n k = (Uα × Wβ) × R ⊕ R .

38 1 n 1 k If (x , . . . , x , y , . . . , y ) =: (x, y) are local coordinates on Eα,β = Uα × Wβ, then we

may put induced local coordinates (x, y, X, Y ) on T (Eα,β) as in Example 2.1.5.

Now the basal and vertical components of a vector (x, y, X, Y ) in T (Eα,β) are given by (x, y, X, 0) and (x, y, 0,Y ), respectively.

Let γ be a path contained in Uα. A lift of γ to Eα,β is path γe : I → Uα × Wβ that is represented in local coordinates by

γe(t) = (γ1(t), . . . , γn(t), γe1(t),..., γek(t)) =: (γ, γe), (4.5.2)

so that π ◦ γe(t) = γ(t) for all t ∈ I. The space of all such lifts is denoted by Γγ(Eα,β). The

velocity lift of any γe ∈ Γγ(Eα,β) is a path in T (Eα,β), given in induced local coordinates by ( ) γe˙ (t) = γ1(t), . . . , γn(t), γe1(t),..., γek(t), γ˙ 1(t),..., γ˙ n(t), γe˙ 1(t),..., γe˙ k(t)

= (γ(t), γe(t), γ˙ (t), γe˙ (t)), (4.5.3)

i dγi e˙ i dγei ∈ e whereγ ˙ (t) = dt (t) and γ (t) = dt (t). In particular, suppose (p, v) = v im(γ). One

can think of the velocity lift of γe as a function from im(γe) ⊂ Eα,β to T (Eα,β). The fiber- ˙ coordinates of γe in Tv(Eα,β) are given by

∂ ∂ γe˙ (v) =γ ˙ i(p) + γe˙ j(v) . (4.5.4) ∂xi ∂yj

The basal components of γe˙ are given by γe˙ B = (γ, γ,e γ,˙ 0), and the vertical components by

γe˙ V = (γ, γ,e 0, γe˙ ).

Evidently, the basal spaces along any lift of γ are completely determined by the original path γ in Uα and do not depend on the particular lift γe at all. Thus, whether or not a lift γe is horizontal is completely determined by the vertical components of its velocity lift.

Recall that H is a global subbundle of TE that is complementary to V , so T (Eα,β) can also be decomposed as Hα,β ⊕Vβ. For p ∈ im(γ), elements of a single horizontal space Hv can be written as direct sums of vectors in Bp and Vv. The basal component is determined by the path γ, and the vertical component is determined by applying the local Christoffel

39 α form Γv to the basal component in each fiber. Continuing to use induced local coordinates as bases for Bα and Vβ, one sees that vectors in Hα,β take the form

∂ ∂ γ˙ i + (Γαγ˙ )j (4.5.5) ∂xi v ∂yj

α along the path γ. Thus, the horizontal spaces along γ have local coordinates (γ, v, γ,˙ Γv γ˙ ). Notice these differ from the cooridnates of the basal space along γ,(γ, v, γ,˙ 0), only by vertical translations, and that these translations are exactly those determined by the local Christoffel forms.

Comparing equation (4.5.4) to the basis of H along γ in (4.5.5), we now have exact criteria for a lift γe of γ to be horizontal.

Proposition 4.5.1 A lift γe ∈ Γγ(Eα,β) is horizontal if and only if

e˙ α α e γ(t) = Γγe(t)(γ ˙ (t)) = Γ (γ(t), γ˙ (t)) (4.5.6) for all t ∈ I. We write γe = γ. □

Equation (4.5.6) gives a system of first order ordinary differential equations for γe along γ.

Given an initial value γe(0) = v ∈ Wβ, the classical FEUT applies, and a unique solution e α γ = γ exists on the part of I for which Γγe is bounded. Invoking Proposition 4.4.5, this is equivalent to the Wong angles θγ being bounded below alongγ ˙ .

40 CHAPTER 5

HORIZONTAL BUNDLES ON TM

A horizontal bundle H on a fiber bundle π : E →→ M describes how elements of the fibers of E change with respect to infinitesimal changes in the base manifold. In case

E = TM, this yields a second-order differential equation, or SODE, associated to H . Del

Riego and Parker [13] use this SODE to study the geometry that a horizontal bundle H induces on the base manifold.

5.1 Vector Bundle Structures

Let M be a manifold with tangent bundle TM. The map π∗ : TTM →→ TM defines a different vector bundle structure on TTM over TM than the standard tangent bundle structure πT : TTM →→ TM.

TTM0  00  00 πT  0π∗  00 × 0 TM TM 00  00  00  π 00  π 0 × M In induced local coordinates, vector addition in each bundle structure is defined by

πT :(x, y, X1,Y1) + (x, y, X2,Y2) = (x, y, X1 + X2,Y1 + Y2), and

π∗ :(x, y1,X,Y1) + (x, y2,X,Y2) = (x, y1 + y2,X,Y1 + Y2).

There is an involution I : TTM → TTM that interchanges the two bundle structures, given by

I(x, y, X, Y ) = (x, X, y, Y ).

It is important to notice that the map I is a fiberwise involution of the total space TTM, but not an involution of V or H . Indeed, if (p, v, 0,V ) ∈ V , then I(p, v, 0,V ) = (p, 0, v, V )

41 is again in V only if v = 0. Similarly, if (p, v, X, Γα(v, X)) ∈ H , then I(p, v, X, Γα(v, X)) =

(p, X, v, Γα(v, X)) is again in H only if Γα(v, X) = Γα(X, v).

Definition 5.1.1 Let H be a horizontal bundle on π : TM →→ M with vertical projection

V. Define the symmetry operator S : X × X → V fiberwise by

S(U, V ) = I · VV (U, 0) − VU (V, 0).

A horizontal bundle H on M is said to be symmetric if and only if its symmetry operator vanishes: S( , ) = 0.

The symmetric horizontal bundles respect both the πT and π∗ structures on TTM simulta- neously. In fact, they determine the same horizontal bundle with respect to each structure.

Notice that such symmetry only makes sense for horizontal subbundles of TTM.

Clearly S(U, V ) = −I · S(V,U), so that S is I-skewsymmetric. Over a trivializing neighbor- hood Uα ⊆ M, S is given by

S(u, v) = I · Vv(p, v, u, 0) − V(p, u, v, 0)

= −I(p, v, u, Γα(v, u)) + (p, u, v, Γα(u, v))

= −(p, u, v, Γα(v, u)) + (p, u, v, Γα(u, v))

= (p, u, 0, Γα(u, v) − Γα(v, u)),

for u, v ∈ TpM. Therefore S is a vertical-vector-valued 2-form on Uα that measures the failure of H to be invariant under the involution I. This can be written as

S(u, v) = Γα(u, v) − Γα(v, u).

5.2 Second-Order Differential Equations

Denote the fixed set of the involution I ∈ Aut(TTM) by S := fix I.A second-order differential equation, or SODE, on M is a section of S . If S ∈ ΓS vanishes on the 0-section

of TM, then S is called a quasispray.

42 The induced tangent map π∗ : TTM → TM is a fiber-wise isomorphism of horizontal ∼ spaces and tangent spaces to M: for each point v ∈ TpM, π∗ : Hv = TpM. This map can be used to define a SODE on M that is compatible with the horizontal bundle H .

Definition 5.2.1 Let H be a horizontal bundle on a smooth manifold M. Then H induces

a horizontal quasispray on M given by

−1 Q(v) = π∗| (v). Hv

We write H ⊢ Q to denote this relationship.

Over a trivializing chart Uα ⊆ M, the quasispray Q can be expressed in local coordinates as

Q(v) = (p, v, v, Γα(v, v)).

Along the 0-section, we obtain

Q(0) = (p, 0, 0, Γα(0, 0)),

α where Γ (0, 0) = 0 since H0 is a vector subspace of T0TM. So Q ⊣ H is indeed a quasispray. It is compatible with H by construction.

5.3 Covariant Derivatives and Connection Geodesics

Suppose now that H is a horizontal bundle on a vector bundle π : E →→ M, and

recall the map K : V → E : Juv 7→ v of Definition 3.1.3. This can be used together with the vertical projection to define a covariant derivative associated to H .

Definition 5.3.1 The covariant derivative associated to a horizontal bundle H is the map

∇ : X × ΓE → ΓE given by

∇X σ = K ◦ V(σ∗X).

for X ∈ X and σ ∈ ΓE. Notice that ∇ is tensorial in X, but merely smooth in σ.

43 Suppose we fix σ ∈ ΓE. Then ∇ σ : X → ΓE is called the covariant differential of σ associated to the horiztonal bundle H along σ. The covariant differential of a horizontal bundle on a vector bundle can be described locally in terms of the local Christoffel forms.

α α ∈ V Example 5.3.2 If E is a vector bundle, then the form Γσ := Γ (σ, ) Γ σ can be composed with the map K : T (Eα) → Eα to give an ΓEα-valued form on Uα,

K ◦ α → Γσ : Xα ΓEα.

K ◦ α ∇ The forms Γσ can be used to give a local description of the covariant differential σ :

Xα → ΓEα. Using local coordinates and some abuse of notation, the covariant differential is given locally by

∇ σ(X) = K ◦ Vσ(σ∗X) ( ) = K ◦ V p, σp,Xp, (σ∗X)p [ ] = K ◦ V (p, σp, 0, (σ∗X)p) + (p, σp,Xp, 0) [ ] K − B − α = (σ∗X α(σ∗X)) Γσ X K ◦ − B − K ◦ α = (1 α)(σ∗X) Γσ (X)

This looks more complicated than it is. The map (1 − Bα) is the projection onto V parallel to Bα. If {ξi} is a frame for the model fiber F and σ is represented locally (over Uα) by

i σ = σ ξi, then it is given by i j ∂σ K(1 − B )(σ∗X) = X ξ . (5.3.1) α ∂xj i

Thus, the covariant differential (or derivative) is given in local coordinates in Eα by

∂σi ∇ σ = Xj ξ − (K ΓαX)iξ . (5.3.2) X ∂xj i σ i

We see that composing the local Christoffel forms with the map K yields the classical con-

nection coefficients [28, 13], or Christoffel symbols [24], for the horizontal bundle H . This

44 justifies naming the maps Γα after Christoffel. Comparing the action of the Christoffel forms

along a fiber of E to the action of the Christoffel symbols within a fiber of E gives the reader

an intuitive idea of how the Christoffel forms actually behave; or more likely, vice versa.

The covariant derivative measures the failure of the section σ to be horizontal along the integral curves of X. If ∇X σ = 0, we say that σ is H -parallel along X, usually shortening this to just parallel when H is understood from context. Thus the covariant derivative ∇X can also be described in terms of certain lifts of the integral curves of X.

Example 5.3.3 Let p ∈ M, and suppose γ : I → M is the unique integral curve of X with

γ(0) = p. Let γ : I → E be the horizontal lift of γ satisfying γ(0) = v, where v = σ(p) ∈ Ep.

Then the covariant derivative ∇X σ along γ is given by [ ] (∇X σ)(p) = K σ˙ (γ) − γ˙ (0). (5.3.3)

Since σ is a section of E, the composition σ(γ) defines an honest lift of γ, and its velocity

lift to TE is well defined. Since σ(γ) and γ are both lifts of γ, they will differ in TvE only by a vertical vector; cf. Section 4.5. Therefore the application of the map K also makes sense.

Clearly, σ is H -parallel along X if and only if σ(γ) = γ for every integral curve of X.

While this is a nice intuitive description of the covariant derivative, it does not appear

to be particularly useful otherwise. The problem is that even if ∇X is evaluated along the

integral curves of X, the horizontal lift γ = Pγ σp must be updated at every point in im(γ).

If E = TM, then ΓE = X and ∇ : X × X → X. In this case, the covariant derivative acts as a directional derivative for vector fields induced by H . Covariant derivatives on

a vector bundle E are in bijective correspondence with horizontal bundles on E [28, 25].

Notice in particular that a horizontal bundle is not required to be a connection in order to

define a covariant derivative.

Now suppose that H is a connection on M. Then for every path γ : I → M, there

is a unique principal horizontal lift of γ that satisfies γ(0) =γ ˙ (0) = v ∈ TpM.

45 Definition 5.3.4 A path is called a geodesic of H , or an H -geodesic, if and only if γ =γ ˙

on the entire domain of γ.

That is,γ is an H -geodesic if and only ifγ ˙ is parallel along γ with respect to H . This is

usually stated in terms of the covariant derivative.

Proposition 5.3.5 A path γ is an H -geodesic if and only if ∇γ˙ γ˙ = 0. □

Since the velocity lift of γ is defined via pullback, one will not encounter any of the problems with self-intersecting paths that could otherwise arise; cf. O’Neill [24, p. 66].

The geodesic equation can be written in terms of the horizontal quasispray H ⊢ Q.

Proposition 5.3.6 A path γ : I → M is an H -geodesic if and only if γ¨ = Q(γ ˙ ).

Notice that this equation makes sense for any horizontal bundle H , and not just those that are connections. This results in constant paths γ(t) = p being geodesics at points where horizontal path lifting breaks down.

Proof: It suffices to prove this for im(γ) contained in a single trivializing chart Uα ⊆ M. In induced local coordinates, the relevant lifts look like

Q(γ ˙ ) = (γ, γ,˙ γ,˙ Γα(γ, ˙ γ˙ )) ,

γ˙ = (γ, γ, γ,˙ Γα(γ, γ˙ )) , and

γ¨ = (γ, γ,˙ γ,˙ γ¨).

A path γ is a geodesic if and only ifγ ˙ = γ, whence the vertical component ofγ ¨ must satisfy

γ¨ = Γα(γ, ˙ γ˙ ). □

Looking only at the vertical component, the geodesic equation can be written locally

in terms of the local Christoffel forms.

α Corollary 5.3.7 A path γ : I → Uα is a (local) H -geodesic if and only if γ¨ = Γ (γ, ˙ γ˙ ). □

46 The quasispray Q can be used to characterize totally geodesic submanifolds of a manifold M.

Proposition 5.3.8 Let ι : N,→ M be a submanifold. Then N is totally geodesic if and only if Q(TN) = TTN.

Proof: A submanifold is totally geodesic if and only if for every point p ∈ N, the geodesics

emanating from p remain in N. Let γv denote the inextendable geodesic emanating from

p with initial velocityγ ˙ (0) = v ∈ TpN. Since γ is a geodesic, thenγ ¨v(0) = Q(v) ∈ TvTN.

Allowing v to vary in TN, we obtain Q(TN) = TTN. □

Equivalently, a submanifold ι : N,→ M is totally geodesic if and only if the horizontal quasispray Q : TM → H is ι∗-related to itself. The following diagram illustrates this situation.

ι∗∗ TTNO / HO

ι∗Q Q

TN ι∗ / TM

πN πM   / N ι M Identifying the pullback ι∗Q with Q, a submanifold ι : N,→ M is totally geodesic with

respect to H if and only if ι∗∗Q = Qι∗.

5.4 Jacobi Fields

Jacobi fields can be defined using only a SODE S on M; in particular, the quasispray

induced by a horizontal bundle H . This follows [25, 14].

Definition 5.4.1 Let c : I → M be an S-geodesic for a SODE S on M.A geodesic variation

of c is a smooth map µ : I × (−ε, ε) → M such that

1. µ(t, 0) = c(t) for all t ∈ I, and

47 2. µ( , s) is an S-geodesic for each s ∈ (−ε, ε).

Definition 5.4.2 Let µ(t, s) be an S-geodesic variation for a SODE S on M with µ(t, 0) = c(t). The variational vector field along c is defined to be V := ∂sµ|s=0. In induced local coordinates, it is a solution of the system

∂Si ∂Si y¨i = (x ˙)y ˙j + (x ˙)y ˙j. ∂xj ∂yj

We refer to this system as the variational equations of the geodesic equation.

It is now possible to define Jacobi fields without needing either a metric tensor or even a connection on M, but simply a SODE.

Definition 5.4.3 A Jacobi field along an S-geodesic γ is an S-geodesic variational vector

field of and along γ.

Jacobi fields can now be used to define conjugate points in the usual way.

Definition 5.4.4 Let γ : [0, b] → M be an S-geodesic segment with γ(0) = p. Then γ(b) is conjugate to p along γ if and only if there exists a nontrivial Jacobi field V along γ such that V (0) = V (b) = 0.

Del Riego and Parker [13] define a family of exponential maps associated to a qua- sispray. It can be shown that the conjugate points along a geodesic are exactly the points where these exponential maps drop rank. This is precisely the case in (pseudo-) Riemannian geometry.

48 CHAPTER 6

UNIFORMLY VERTICALLY BOUNDED HORIZONTAL BUNDLES

In this chapter, we give the first complete answer to the following question:

Which horizontal bundles are Ehresmann connections?

In particular, we give exact criteria for a horizontal bundle to admit a system of parallel

transport P in the total space of a bundle E, along the base manifold M.

6.1 Uniform Vertical Boundedness

In chapter 5 we saw that every horizontal bundle on TM determines a second-order differential equation on the base manifold M. In case E is an arbitrary fiber bundle, a horizontal bundle H doesn’t define an honest SODE, but it still acts analogously on the

fibers of E. That is, a horizontal bundle on E is a system of first-order differential equations on E, whose integral manifolds are transverse to the fibers of E, if they exist. To ensure that the integral manifolds intersect each fiber of E, it is necessary to bound the horizontal spaces away from the vertical spaces.

Definition 6.1.1 A horizontal bundle H on a fiber bundle π : E →→ M is said to be uniformly vertically bounded (UVB) if and only if the horizontal spaces Hv are bounded away from the vertical spaces Vv, uniformly along fibers of E.

This condition can be formulated in terms of Wong angles.

Lemma 6.1.2 A horizontal bundle H on a fiber bundle π : E →→ M is uniformly vertically bounded if and only if for every local trivialization Eα = Uα ×F , there exists a non-vanishing function ε ∈ F(Uα) such that θp ≥ εp > 0 for all p ∈ Uα, where θp is the lower bound for the

Wong angles along Ep defined in equation (4.3.1).

Proof: The Wong angles measure the distance between the vertical space Vv and the horizontal space Hv in a single fiber TvE. Thus a horizontal bundle H is bounded away

49 from V , uniformly along a fiber Ep, if and only if the Wong angles are bounded away from

0 along Ep. This is exactly what the lower bound function θp measures. The function

ε ∈ F(Uα) must be smooth since H is smooth. □

Proposition 4.4.5 immediately implies the following corollary.

Corollary 6.1.3 Let H be a horizontal bundle on a smooth fiber bundle π : E →→ M. Then

H is UVB if and only if the local Christoffel forms Γα are bounded along fibers of E. □

Example 6.1.4 Let H be a linear connection on a vector bundle π : E →→ M, and recall

that linear connections satisfy Hav = a∗Hv. The action of a∗ is a motion of TvE, so therefore

preserves the Wong angles along the line through 0 in Ep determined by v [34, 4]. Hence the

Wong angles are constant along each fiber Ep so there is an absolute minimum value among

m them, say θ > 0, and this is uniform along the fiber Ep. Therefore all linear connections are uniformly vertically bounded.

It is easy to see that the Wong angles being constant along each fiber of a vector

bundle E, or vertically constant, characterizes linear connections.

Theorem 6.1.5 A horizontal bundle H on a vector bundle E is a linear connection if and

only if its Wong angles are vertically constant. □

The notion of a horizontal bundle with vertically constant Wong angles extends to

smooth fiber bundles.

Definition 6.1.6 A horizontal bundle H on a fiber bundle π : E →→ M is said to be

vertically constant if and only if its Wong angles are constant along fibers of E.

A vertically constant horizontal bundle is completely determined by its Wong angles

Θ along any section of E, if a global section exists. Obviously every vertically constant horizontal bundle is UVB, but we note it here for completeness.

Proposition 6.1.7 Every vertically constant horizontal bundle is UVB. □

50 6.2 HPL if and only if UVB

Recall that parallel transport is defined via horizontal lifts of paths from M to E.

A horizontal bundle H determines a system of parallel transport in E if and only if H has horizontal path lifting (HPL) (Definition 3.3.4). Uniform vertical boundedness (UVB) is exactly what is needed to ensure HPL, hence also the existence of parallel transport.

Theorem 6.2.1 A horizontal bundle H on a fiber bundle π : E →→ M has horizontal path lifting (HPL) if and only if it is uniformly vertically bounded (UVB).

Proof: Let γ : I → M be a path with γ(0) = p and γ(1) = q. Without loss of generality, we may assume that im(γ) lies in a single chart Uα that is trivializing for both E and TM. Indeed, since I = [0, 1] is compact, then im(γ) is compact in M. Thus im(γ) can be covered by finitely many such charts. We may then partition im(γ) into finitely many subpaths, each lying in a single chart, and reparametrize each one so that its domain is I = [0, 1].

Let γe = γ♮ c : I → Eα be a lift of γ, and similarly suppose that there is a chart

Wβ ⊆ F that is trivializing for TF such that im(γe) ⊂ Uα × Wβ. Again, since I is compact, then im(γe) is compact in Eα = Uα × F . We may thus cover im(γe) by finitely many charts of the form Uα × Wβ, and partition im(γe) as we did for γ. We may further suppose that the projection of im(γe) to Wβ has no self intersections. Indeed, if it does, then partition im(γe) further to remove them.

It is now possible to consider im(γe) in local coordinates as in (4.5.2). Proposition

˙ α 4.5.1 implies that γe ∈ Γγ(Eα,β) is horizontal if and only if it satisfies γe(t) = Γ (γ ˙ (t), γe(t)) for all t ∈ I. This yields a system of first-order ordinary differential equations for γe along

γ. Since we have reduced the problem to local coordinates, standard ODE theory applies.

Recall that the lift γe = (γ, γe) will exist over all of I if and only if γe : I → γ∗E is defined on all of I.

Since H is UVB, Corollary 6.1.3 implies that ∥Γα∥ is uniformly bounded above. A

Mean Value Theorem [3, p. 366] implies that ∥γe∥ is bounded on the part of I where it exists.

51 The Fundamental Existence and Uniqueness Theorem [20, pp. 166f, 169] gives the existence of γe, and the Extension Theorem [20, p. 171f ] implies that γe extends over all of I.

Conversely, if H has HPL, then the horizontal foliation Pγ of Proposition 3.3.3 is

∗ such that every leaf intersects every fiber of γ E. Since the leaves of Pγ are integral curves of H , this implies that H is UVB. □

Theorem 6.2.1 was first proved by Parker for the special case when E = TM. It was then extended to vector bundles, and finally to smooth fiber bundles. The vector bundle and tangent bundle proofs can be thought of as more simple-minded as they take advantage of the algebraic structure on the fibers of E. However, they are less revealing as to the underlying topological relationship between a horizontal bundle and the vertical bundle.

One significant advantage that vector bundles have is that vector spaces are paralleliz- able: if F is an n-dimensional vector space, then TF = F × Rn. This allows one to greatly simplify the content of Section 4.3. In particular, for the complete Riemannian metric gF we may use the standard Euclidean metric on F . This is the preferred choice of metric on F because the parallel transport it induces is trivial: all of the diffeomorphisms Pγ are the identity map. Thus, we may compare Wong angles along fibers Ep without needing to explicitly parallel transport the spaces Hw and Vw to sit over a common base point v ∈ Ep.

The lower bound along the fiber Ep is then simply

θp := inf θv. (6.2.1) v∈Ep Tangent bundles have even further advantages. For every pair (p, v) ∈ TM, the spaces TpM, Vv = Tv(TpM), and Hv are all n-dimensional vector spaces. This allows one to identify them with each other ad libitum over a trivializing chart Uα.

The original proofs of these special cases are now included to illustrate the differences between tangent bundles, vector bundles, and arbitrary smooth fiber bundles.

Corollary 6.2.2 Let H be a horizontal bundle on a vector bundle π : E →→ M; then H has HPL if and only if H is UVB.

52 Proof: Let γ : I → M be a path in M with γ(0) = p and γ(1) = q. Without loss of generality, we may assume that im(γ) is contained in a single trivializing chart (Uα, φα) for

k both E and TM. Restricted to this chart, the total space Eα looks like Eα = Uα × R .

∗ k Let c ∈ Γ(γ E) be a smooth curve such that c(0) = v ∈ R and cot c(t) ∈ Hc(t) for all t ∈ I. In induced local coordinates,c ˙(t) = (γ(t), c(t),Dnγ(t),Dkc(t)), where c(t) is the Eγ(t)

fiber-component, Dnγ(t) is the basal component, and Dkc(t) is the vertical component.

The UVB property implies that both ∥Dnγ∥ and ∥Dkc∥ are bounded above, uniformly along the fibers of γ∗E. Identifying the fibers of γ∗E over I, these bounds are uniform in

n k R and R in the usual sense. The basal component Dnγ will play no further role, as it is completely determined by the path γ.

The remainder of the proof is an exercise in analysis. An MVT [3, p. 366] implies that since ∥Dkc∥ is uniformly bounded, then ∥c∥ is bounded on the part of I where it exists. The FEUT [20, pp. 162f,169] provides the existence of c, and the Extension Theorem [20, p. 171f ] shows that c extends to all of I.

∗ By definition, c ∈ Γ(γ E) is a horizontal section. Therefore γ = γ♮c is a horizontal section of E along γ with γ(0) = v and γ(1) ∈ Eq. Uniqueness of γ follows from that of c.

Conversely, since c is smooth, then ∥c∥ bounded implies that ∥Dkc∥ is bounded. The rest of this direction follows similarly. □

Corollary 6.2.3 A horizontal bundle H on TM has HPL if and only if it is UVB.

This is Parker’s original proof [27, v1].

∗ ∗ ∼ × Rn Proof: Consider the pullback bundle γ TM over I. Note that γ TM = I T . The label Rn ∗ ∼ R2n Rn ⊕ Rn will be used to distinguish copies of . The pullback γ TTM = = B V . The H ∈ Rn horizontal bundle pulls back to the family of horizontal n-planes u for each u T . We Rn Rn may assume that V is the vertical space, and shall refer to B as the basal space. → Rn ∈ Rn ∈ H We seek a curve c : I T such that c(0) = v T andc ˙(t) c(t). It is Rn Rn Rn convenient to identify T and B. Let D denote the usual derivative in so that Dc is

53 ∈ Rn the Jacobian matrix of c. Then with c(t) B, we havec ˙(t) = (c(t), Dc(t)); i.e., c(t) is the basal component and Dc(t) is the vertical component ofc ˙(t).

Now the UVB condition implies the existence of an upper bound on ∥Dc∥ which is

uniform along the fibers of γ∗TM. With I compact, the bound may also be taken to be

uniform along I. Applying an MVT [3, p. 366], this implies that ∥c∥ is bounded on the

part of I where it exists. The FEUT [20, pp. 162f,169] provides the existence of c, and the

Extension Theorem [20, p. 171f ] shows that c extends to all of I.

So we may regard c ∈ Γ(γ∗TM) over I. By definition, c is a horizontal section. It

follows that the pushforward γ of c is a horizontal section of TM along γ with γ(0) = v and

γ(1) ∈ TqM. Uniqueness of γ follows immediately from that of c. The converse follows similarly, noting that ∥c∥ bounded implies ∥Dc∥ is also bounded since c is smooth. □

Combining the results of this section, we obtain the following theorem.

Theorem 6.2.4 Let H be a horizontal bundle on a smooth fiber bundle π : E →→ M with

model fiber F . The following are equivalent.

i.) H is uniformly vertically bounded;

ii.) H is an Ehresmann connection; i.e., H has horizontal path lifting;

iii.) H admits a system of parallel transport in E along M;

iv.) There exists a Riemannian metric on F such that the Christoffel forms Γα are bounded

above, along the fibers of E. □

Kol´aˇr,Michor, and Slov´ak[22, p. 81] state a result equivalent to iv.) ⇒ ii.). An explicit

proof is not given, but one may be inferred from the surrounding text. Nonetheless, mention

is never made of anything akin to Wong angles in their text, so we may assume that their

point of view was at least slightly different.

If E = TM, then the horizontal quasispray may be included in this equivalence.

54 Corollary 6.2.5 A horizontal bundle H on a tangent bundle π : TM →→ M has HPL if and only if its induced quasispray is bounded along fibers of TM.

Proof: The quasispray Q ⊣ H is given in induced local coordinates by

Q(v) = (p, v, v, Γα(v, v)).

Therefore Q is bounded along fibers of TM if and only if Γα(v, v) is bounded along fibers of

TM. □

The following proposition was proved by Ehresmann in [17] using completely different methods. It is easy to prove using Wong angles.

Proposition 6.2.6 Let π : E →→ M be a fiber bundle with compact model fiber F . Then every horizontal bundle on E is a connection.

Proof: The Wong angles θv in any fiber of E can never equal 0 since Hv must be comple- mentary to Vv for every v ∈ E. Since the fibers Ep are compact, then θp must be bounded away from 0 for every p ∈ M. □

Finally, with this new perspective we see that the proof of Theorem 3.2.2 actually proves that every fiber bundle π : E →→ M admits a horizontal bundle with HPL.

Theorem 6.2.7 Every smooth fiber bundle π : E →→ M admits a connection.

Proof: Let g be an auxiliary Riemannian metric on the total space E, and let Hv = Nv(Ep) for every v ∈ Ep and all p ∈ M. Theorem 3.2.2 implies that this H is a horizontal bundle. 1 ··· n ∈ The Wong angles in TvE are given by θv = = θv = π/2 for every v E, as measured by the metric g. Therefore H is UVB. □

55 CHAPTER 7

CONCLUSION AND FUTURE DIRECTION

The primary goal of this research project was to give a geometric characterization of the horizontal bundles over a fiber bundle which are actually connections. In doing so, it became clear that while HPL is necessary to define a notion of parallel transport in a fiber bundle, many other geometric properties that are commonly attributed to a connection do not require HPL. In particular, every horizontal bundle on TM defines both a covariant derivative and a quasispray. The quasispray can then be used to define geodesics of the horizontal bundle. Many geometric properties then follow without requiring the horizontal bundle to be a connection. Similarly, every horizontal bundle defines a notion of curvature.

This revelation begs many questions to be answered in future work. Specifically, what other geometric properties of a fiber bundle do, or do not, depend on a horizontal bundle having HPL? Which properties come specifically from the quasispray, and are not even specific to a horizontal bundle? The Wong angles and local Christoffel forms should be helpful in attempting to answer these types of questions.

Moreover, the space of horizontal bundles over a given fiber bundle π : E →→ M should be studied in more depth. The space of connections on E is a subspace of the space of horizontal bundles. Can anything meaningful be said about it? For example, what kind of subspace is it? Are there any interesting algebraic topological invariants?

56 REFERENCES

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