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Geometry of Horizontal Bundles and Connections A 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.
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