Differential Geometry: Connections, Curvature, and

Loring W. Tu

Differential Geometry: Connections, Curvature, and Characteristic Classes April 8, 2017

Springer Berlin Heidelberg New York Hong Kong London Milan Paris Tokyo

Loring W. Tu Department of Mathematics Tufts University Medford, MA 02155 [email protected]

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Loring W. Tu 2017

Preface

Differential geometry has a long and glorious history. As its name implies, it is the study of geometry using differential calculus, and as such, it dates back to Newton and Leibniz in the seventeenth century. But it was not until the nineteenth century, with the work of Gauss on surfaces and Riemann on the curvature tensor, that differential geometry flourished and its modern foundation was laid. Over the past one hundred years, differential geometry has proven indispensable to an understanding of the physical world, in Einstein’s general theory of relativity, in the theory of gravitation, in gauge theory, and now in string theory. Differential geometry is also useful in topology, several complex variables, algebraic geometry, complex manifolds, and dynamical systems, among other fields. It has even found applications to group theory as in Gromov’s work and to probability theory as in Diaconis’s work. It is not too far-fetched to argue that differential geometry should be in every mathematician’s arsenal. The basic objects in differential geometry are manifolds endowed with a metric, which is essentially a way of measuring the length of vectors. A metric gives rise to the notions of distance, angle, area, volume, curvature, straightness, and geodesics. It is the presence of a metric that distinguishes geometry from topology. However, another concept that might contest the primacy of a metric in differential geometry is that of a connection. A connection in a vector bundle may be thought of as a way of differentiating sections of the vector bundle. A metric determines a unique connection called a Riemannian connection with certain desirable properties. While a connection is not as intuitive as a metric, it already gives rise to curvature and geodesics. With this, the connection can also lay claim to be a fundamental notion of differential geometry. Indeed, in 1989, the great geometer S. S. Chern wrote as the editor of a volume on global differential geometry [5], “The Editor is convinced that the notion of a connection in a vector bundle will soon find its way into a class on advanced calculus, as it is a fundamental notion and its applications are wide-spread.” In 1977, the Nobel-prize winning physicist C. N. Yang wrote in [23], “Gauge fields are deeply related to some profoundly beautiful ideas of contemporary mathematics, ideas that are the driving forces of part of the mathematics of the last 40

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years, ... , the theory of fiber bundles.” Convinced that gauge fields are related to connections on fiber bundles, he tried to learn fiber-bundle theory from several mathematical classics on the subject, but “learned nothing. The language of modern mathematics is too cold and abstract for a physicist” [24, p. 73]. While the definition and formal properties of a connection on a principal bundle can be given in a few pages, it is difficult to understand its meaning without knowing how it came into being. The present book is an introduction to differential geometry that follows the historical development of the concepts of connection and curvature, with the goal of explaining the Chern–Weil theory of characteristic classes on a principal bundle. The goal, once fixed, dictates the choice of topics. Starting with directional derivatives in an Euclidean space, we introduce and successively generalize connections and curvature from a tangent bundle to a vector bundle and finally to a principal bundle. Along the way, the narrative provides a panorama of some of the high points in the history of differential geometry, for example, Gauss’ Theorema Egregium and the Gauss–Bonnet theorem. Initially, the prerequisites are minimal; a passing acquaintance with manifolds suffices. Starting with Section 11, it becomes necessary to understand and be able to manipulate differential forms. Beyond Section 22, a knowledge of de Rham cohomology is required. All of this is contained in my book An Introduction to Manifolds [21], and can be learned in one semester. It is my fervent hope that the present book will be accessible to physicists as well as mathematicians. For the benefit of the reader and to establish common notations, we recall in Appendix A the basics of manifold theory. In an attempt to make the exposition more self-contained, I have also included sections on algebraic constructions such as the tensor product and the exterior power. In two decades of teaching from this manuscript, I have generally been able to cover the first twenty-five sections in one semester, assuming a one-semester course on manifolds as the prerequisite. By judiciously leaving some of the sections as independent reading material, for example, Sections 9, 15, and 26, I have been able to cover the first thirty sections in one semester. Every book reflects the biases and interests of its author. This book is no exception. For a different perspective, the reader may find it profitable to consult other books. After having read this one, it should be easier to read the others. There are many good books on differential geometry, each with its particular emphasis. Some of the ones I have liked include Boothby [1], Conlon [6], do Carmo [7], Kobayashi and Nomizu [12], Lee [14], Millman and Parker [16], Spivak [19], and Taubes [20]. For applications to physics, see Frankel [9]. As a student, I attended many lectures of Phillip A. Griffiths and Raoul Bott on algebraic and differential geometry. It is a pleasure to acknowledge their influence. I want to thank Andreas Arvanitoyeorgos, Jeffrey D. Carlson, Benoit Charbonneau, Hanci Chi, Brendan Foley, George Leger, Shibo Liu, Ishan Mata, Steven Scott, and Huaiyu Zhang for their careful proofreading, useful comments, and errata lists. Jeffrey D. Carlson in particular should be singled out for the many excellent pieces of advice he has given me over the years. I also want to thank Bruce Boghosian for helping me with Mathematica and for preparing the figure of the Frenet–Serret frame

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(Figure 2.5). Finally, I am grateful to the Max Planck Institute for Mathematics in Bonn, National Taiwan University, and the National Center for Theoretical Sciences in Taipei for hosting me during the the preparation of this manuscript.

Medford, Massachusetts April 2017

Loring W. Tu

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

Frontispiece . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Chapter 1 Curvature and Vector Fields §1

Riemannian Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Inner Products on a Vector Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Representations of Inner Products by Symmetric Matrices . . . . . . 1.3 Riemannian Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Existence of a Riemannian Metric . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 4 5 6 8 9

§2

Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Regular Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Arc Length Parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Signed Curvature of a Plane Curve . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Orientation and Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 12 13 15 16

§3

Surfaces in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Principal, Mean, and Gaussian Curvatures . . . . . . . . . . . . . . . . . . . . 3.2 Gauss’s Theorema Egregium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Gauss–Bonnet Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19 19 21 22 23

§4

Directional Derivatives in Euclidean Space . . . . . . . . . . . . . . . . . . . . . . . 4.1 Directional Derivatives in Euclidean Space . . . . . . . . . . . . . . . . . . . 4.2 Other Properties of the Directional Derivative . . . . . . . . . . . . . . . . . 4.3 Vector Fields Along a Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Vector Fields Along a Submanifold . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Directional Derivatives on a Submanifold of Rn . . . . . . . . . . . . . . .

24 24 26 27 28 29

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Contents

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

§5

The Shape Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Normal Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Shape Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Curvature and the Shape Operator . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 The First and Second Fundamental Forms . . . . . . . . . . . . . . . . . . . . 5.5 The Catenoid and the Helicoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 31 32 34 37 38 41

§6

Affine Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Affine Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Torsion and Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 The Riemannian Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Orthogonal Projection on a Surface in R3 . . . . . . . . . . . . . . . . . . . . 6.5 The Riemannian Connection on a Surface in R3 . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 45 46 47 48 49 50

§7

Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Definition of a Vector Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The Vector Space of Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Extending a Local Section to a Global Section . . . . . . . . . . . . . . . . 7.4 Local Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Restriction of a Local Operator to an Open Subset . . . . . . . . . . . . . 7.6 Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 F-Linearity and Bundle Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Multilinear Maps over Smooth Functions . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 51 53 54 55 56 58 58 61 61

§8

Gauss’s Theorema Egregium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 The Gauss and Codazzi–Mainardi Equations . . . . . . . . . . . . . . . . . 8.2 A Proof of the Theorema Egregium . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 The Gaussian Curvature in Terms of an Arbitrary Basis . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63 63 65 66 66

§9

Generalizations to Hypersurfaces in Rn+1 . . . . . . . . . . . . . . . . . . . . . . . . 9.1 The Shape Operator of a Hypersurface. . . . . . . . . . . . . . . . . . . . . . . 9.2 The Riemannian Connection of a Hypersurface . . . . . . . . . . . . . . . 9.3 The Second Fundamental Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 The Gauss Curvature and Codazzi–Mainardi Equations . . . . . . . .

68 68 69 70 70

Chapter 2 Curvature and Differential Forms §10 Connections on a Vector Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Connections on a Vector Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Existence of a Connection on a Vector Bundle . . . . . . . . . . . . . . . . 10.3 Curvature of a Connection on a Vector Bundle . . . . . . . . . . . . . . . . 10.4 Riemannian Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73 74 75 76 76

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10.5 Metric Connections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Restricting a Connection to an Open Subset . . . . . . . . . . . . . . . . . . 10.7 Connections at a Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77 78 79 80

§11 Connection, Curvature, and Torsion Forms . . . . . . . . . . . . . . . . . . . . . . . 11.1 Connection and Curvature Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Connections on a Framed Open Set . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 The Gram–Schmidt Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Metric Connection Relative to an Orthonormal Frame . . . . . . . . . . 11.5 Connections on the Tangent Bundle . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81 81 83 83 84 86 88

§12 The Theorema Egregium Using Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 The Gauss Curvature Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 The Theorema Egregium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Skew-Symmetries of the Curvature Tensor . . . . . . . . . . . . . . . . . . . 12.4 Sectional Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Poincaré Half-Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90 90 92 93 94 94 96

Chapter 3 Geodesics §13 More on Affine Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 13.1 Covariant Differentiation Along a Curve . . . . . . . . . . . . . . . . . . . . . 97 13.2 Connection-Preserving Diffeomorphisms . . . . . . . . . . . . . . . . . . . . 100 13.3 Christoffel Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 §14 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 The Definition of a Geodesic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Reparametrization of a Geodesic . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Existence of Geodesics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Geodesics in the Poincaré Half-Plane . . . . . . . . . . . . . . . . . . . . . . . . 14.5 Parallel Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6 Existence of Parallel Translation Along a Curve . . . . . . . . . . . . . . . 14.7 Parallel Translation on a Riemannian Manifold . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

105 105 107 108 110 112 113 114 115

§15 Exponential Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 The Exponential Map of a Connection . . . . . . . . . . . . . . . . . . . . . . . 15.2 The Differential of the Exponential Map . . . . . . . . . . . . . . . . . . . . . 15.3 Normal Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4 Left-Invariant Vector Fields on a Lie Group . . . . . . . . . . . . . . . . . . 15.5 Exponential Map for a Lie Group . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6 Naturality of the Exponential Map for a Lie Group . . . . . . . . . . . . 15.7 Adjoint Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

117 117 119 120 121 122 124 125

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15.8 Associativity of a Bi-Invariant Metric on a Lie Group . . . . . . . . . . 126 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 15.9

Addendum. The Exponential Map as a Natural Transformation . . 129

§16 Distance and Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1 Distance in a Riemannian Manifold . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Geodesic Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3 Dual 1-Forms Under a Change of Frame . . . . . . . . . . . . . . . . . . . . . 16.4 Volume Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5 The Volume Form in Local Coordinates. . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

131 131 133 134 135 137 138

§17 The Gauss–Bonnet Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1 Geodesic Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 The Angle Function Along a Curve . . . . . . . . . . . . . . . . . . . . . . . . . 17.3 Signed Geodesic Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4 Gauss–Bonnet Formula for a Polygon . . . . . . . . . . . . . . . . . . . . . . . 17.5 Triangles on a Riemannian 2-Manifold . . . . . . . . . . . . . . . . . . . . . . 17.6 Gauss–Bonnet Theorem for a Surface . . . . . . . . . . . . . . . . . . . . . . . 17.7 Gauss–Bonnet Theorem for a Hypersurface in R2n+1 . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

141 141 142 142 145 147 148 150 150

Chapter 4 Tools from Algebra and Topology §18 The Tensor Product and the Dual Module . . . . . . . . . . . . . . . . . . . . . . . . 18.1 Construction of the Tensor Product . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 Universal Mapping Property for Bilinear Maps . . . . . . . . . . . . . . . . 18.3 Characterization of the Tensor Product . . . . . . . . . . . . . . . . . . . . . . 18.4 A Basis for the Tensor Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.5 The Dual Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.6 Identities for the Tensor Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.7 Functoriality of the Tensor Product . . . . . . . . . . . . . . . . . . . . . . . . . . 18.8 Generalization to Multilinear Maps . . . . . . . . . . . . . . . . . . . . . . . . . 18.9 Associativity of the Tensor Product . . . . . . . . . . . . . . . . . . . . . . . . . 18.10 The Tensor Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

155 156 157 158 160 161 162 164 165 165 166 167

§19 The Exterior Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.1 The Exterior Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2 Properties of the Wedge Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3 Universal Mapping Property for Alternating k-Linear Maps . . . . . V 19.4 A Basis for k V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.5 Nondegenerate Pairings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V V 19.6 A Nondegenerate Pairing of k (V ∨ ) with k V . . . . . . . . . . . . . . . 19.7 A Formula for the Wedge Product . . . . . . . . . . . . . . . . . . . . . . . . . .

168 168 168 170 171 173 174 176

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Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 §20 Operations on Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.1 Vector Subbundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2 Subbundle Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3 Quotient Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4 The Pullback Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.5 Examples of the Pullback Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.6 The Direct Sum of Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . 20.7 Other Operations on Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

178 178 179 180 181 184 185 187 189

§21 Vector-Valued Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1 Vector-Valued Forms as Sections of a Vector Bundle . . . . . . . . . . . 21.2 Products of Vector-Valued Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Directional Derivative of a Vector-Valued Function . . . . . . . . . . . . 21.4 Exterior Derivative of a Vector-Valued Form . . . . . . . . . . . . . . . . . . 21.5 Differential Forms with Values in a Lie Algebra . . . . . . . . . . . . . . . 21.6 Pullback of Vector-Valued Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.7 Forms with Values in a Vector Bundle . . . . . . . . . . . . . . . . . . . . . . . 21.8 Tensor Fields on a Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.9 The Tensor Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.10 Remark on Signs Concerning Vector-Valued Forms . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

190 190 192 194 194 195 197 198 199 200 201 201

Chapter 5 Vector Bundles and Characteristic Classes §22 Connections and Curvature Again . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.1 Connection and Curvature Matrices Under a Change of Frame . . . 22.2 Bianchi Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 The First Bianchi Identity in Vector Form . . . . . . . . . . . . . . . . . . . . 22.4 Symmetry Properties of the Curvature Tensor . . . . . . . . . . . . . . . . . 22.5 Covariant Derivative of Tensor Fields . . . . . . . . . . . . . . . . . . . . . . . 22.6 The Second Bianchi Identity in Vector Form . . . . . . . . . . . . . . . . . . 22.7 Ricci Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.8 Scalar Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.9 Defining a Connection Using Connection Matrices . . . . . . . . . . . . 22.10 Induced Connection on a Pullback Bundle . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

204 205 207 208 209 210 211 212 213 213 214 215

§23 Characteristic Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.1 Invariant Polynomials on gl(r, R) . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.2 The Chern–Weil Homomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.3 Characteristic Forms are Closed . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.4 Differential Forms Depending on a Real Parameter . . . . . . . . . . . . 23.5 Independence of Characteristic Classes of a Connection . . . . . . . .

216 216 217 219 220 222

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23.6 Functorial Definition of a Characteristic Class . . . . . . . . . . . . . . . . 224 23.7 Naturality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 §24 Pontrjagin Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.1 Vanishing of Characteristic Classes . . . . . . . . . . . . . . . . . . . . . . . . . 24.2 Pontrjagin Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.3 The Whitney Product Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §25 The Euler Class and Chern Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.1 Orientation on a Vector Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.2 Characteristic Classes of an Oriented Vector Bundle . . . . . . . . . . . 25.3 The Pfaffian of a Skew-Symmetric Matrix . . . . . . . . . . . . . . . . . . . . 25.4 The Euler Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.5 Generalized Gauss–Bonnet Theorem . . . . . . . . . . . . . . . . . . . . . . . . 25.6 Hermitian Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.7 Connections and Curvature on a Complex Vector Bundle . . . . . . . 25.8 Chern Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

227 227 229 230 232 232 233 234 237 237 238 238 239 239

§26 Some Applications of Characteristic Classes . . . . . . . . . . . . . . . . . . . . . . 26.1 The Generalized Gauss–Bonnet Theorem . . . . . . . . . . . . . . . . . . . . 26.2 Characteristic Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.3 The Cobordism Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.4 The Embedding Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.5 The Hirzebruch Signature Formula . . . . . . . . . . . . . . . . . . . . . . . . . . 26.6 The Riemann–Roch Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

241 241 241 242 242 243 243

Chapter 6 Principal Bundles and Characteristic Classes §27 Principal Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.1 Principal Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.2 The Frame Bundle of a Vector Bundle . . . . . . . . . . . . . . . . . . . . . . . 27.3 Fundamental Vector Fields of a Right Action . . . . . . . . . . . . . . . . . 27.4 Integral Curves of a Fundamental Vector Field . . . . . . . . . . . . . . . . 27.5 Vertical Subbundle of the Tangent Bundle T P . . . . . . . . . . . . . . . . . 27.6 Horizontal Distributions on a Principal Bundle . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

245 246 250 251 253 254 255 256

§28 Connections on a Principal Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.1 Connections on a Principal Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . 28.2 Vertical and Horizontal Components of a Tangent Vector . . . . . . . 28.3 The Horizontal Distribution of an Ehresmann Connection. . . . . . . 28.4 Horizontal Lift of a Vector Field to a Principal Bundle . . . . . . . . . 28.5 Lie Bracket of a Fundamental Vector Field . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

258 258 260 261 263 264 264

§29 Horizontal Distributions on a Frame Bundle . . . . . . . . . . . . . . . . . . . . . . 266

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29.1 Parallel Translation in a Vector Bundle . . . . . . . . . . . . . . . . . . . . . . 29.2 Horizontal Vectors on a Frame Bundle . . . . . . . . . . . . . . . . . . . . . . . 29.3 Horizontal Lift of a Vector Field to a Frame Bundle . . . . . . . . . . . . 29.4 Pullback of a Connection on a Frame Bundle Under a Section . . . §30 Curvature on a Principal Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.1 Curvature Form on a Principal Bundle . . . . . . . . . . . . . . . . . . . . . . . 30.2 Properties of the Curvature Form . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

266 268 270 272 274 274 275 278

§31 Covariant Derivative on a Principal Bundle . . . . . . . . . . . . . . . . . . . . . . . 31.1 The Associated Bundle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 The Fiber of the Associated Bundle . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Tensorial Forms on a Principal Bundle . . . . . . . . . . . . . . . . . . . . . . . 31.4 Covariant Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 A Formula for the Covariant Derivative of a Tensorial Form . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

279 279 280 281 284 286 290

§32 Characteristic Classes of Principal Bundles . . . . . . . . . . . . . . . . . . . . . . . 32.1 Invariant Polynomials on a Lie Algebra . . . . . . . . . . . . . . . . . . . . . . 32.2 The Chern–Weil Homomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

291 291 291 295

Appendix §A

Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Manifolds and Smooth Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Tangent Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5 Exterior Differentiation on a Manifold . . . . . . . . . . . . . . . . . . . . . . . A.6 Exterior Differentiation on R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.7 Pullback of Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

297 297 299 300 301 303 306 307 308

§B

Invariant Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1 Polynomials Versus Polynomial Functions . . . . . . . . . . . . . . . . . . . B.2 Polynomial Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3 Invariant Polynomials on gl(r, F) . . . . . . . . . . . . . . . . . . . . . . . . . . . B.4 Invariant Complex Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.5 L-Polynomials, Todd Polynomials, and the Chern Character . . . . . B.6 Invariant Real Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.7 Newton’s Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

310 310 311 312 314 317 319 321 323

Hints and Solutions to Selected End-of-Section Problems . . . . . . . . . . . . . . . 325

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List of Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

A Circle Bundle over a Circle by Lun-Yi Tsai and Zachary Treisman, 2010. Welded stainless steel and latex sheeting. Printed with permission of Lun-Yi Tsai.

Chapter 1 Curvature and Vector Fields

By a manifold, we will always mean a smooth manifold. To understand this book, it is helpful to have had some prior exposure to the theory of manifolds. The reference [21] contains all the background needed. For the benefit of the reader, we review in Appendix A, mostly without proofs, some of the definitions and basic properties of manifolds. Appendix A concerns smooth maps, differentials, vector fields, and differential forms on a manifold. These are part of the differential topology of manifolds. The focus of this book is instead on the differential geometry of manifolds. Now a manifold will be endowed with an additional structure called a Riemannian metric, which gives a way of measuring length. In differential geometry, the notions of length, distance, angles, area, and volume make sense, whereas in differential topology, since a manifold can be stretched and still be diffeomorphic to the original, these concepts obviously do not make sense. Some of the central problems in differential geBernhard Riemann ometry originate in everyday life. Consider the problem in cartography of representing the surface of the (1826–1866) earth on a flat piece of paper. A good map should show accurately distances between any two points. Experience suggests that this is not possible on a large scale. We are all familiar with the Mercator projection which vastly distorts countries near the north and south poles. On a small scale there are fairly good maps, but are they merely approximations or can there be truly accurate maps in a mathematical sense? In other words, is there a distance-preserving bijection from an open subset of the sphere to some open subset of the plane? Such a map is an isometry. Isometry is also related to a problem in industrial design. Certain shapes such as circular cylinders and cones are easy to manufacture because they can be obtained from a flat sheet by bending. If we take a sheet of paper and bend it in various ways,

4

§1 Riemannian Manifolds

we obtain infinitely many surfaces in space, and yet none of them appear to be a portion of a sphere or an ellipsoid. Which shapes can be obtained from one another by bending? In 1827 Carl Friedrich Gauss laid the foundation for the differential geometry of surfaces in his work Disquisitiones generales circa superficies curvas (General investigation of curved surfaces). One of his great achievements was the proof of the invariance of Gaussian curvature under distance-preserving maps. This result is known as Gauss’s Theorema Egregium, which means “remarkable theorem” in Latin. By the Theorema Egregium, one can use the Gaussian curvature to distinguish non-isometric surfaces. In the first eight sections of this book, our goal is to introduce enough basic constructions of differential geometry to prove the Theorema Egregium.

§1 Riemannian Manifolds A Riemannian metric is essentially a smoothly varying inner product on the tangent space at each point of a manifold. In this section we recall some generalities about an inner product on a vector space and by means of a partition of unity argument, prove the existence of a Riemannian metric on any manifold.

1.1 Inner Products on a Vector Space A point u in R3 will denote either an ordered triple (u1 , u2 , u3 ) of real numbers or a column vector  1 u  u2  . u3 The Euclidean inner product, or the dot product, on R3 is defined by 3

hu, vi = ∑ ui vi . i=1

In terms of this, one can define the length of a vector p kvk = hv, vi,

(1.1)

the angle θ between two nonzero vectors (Figure 1.1) cos θ =

hu, vi , kukkvk

0 ≤ θ ≤ π,

and the arc length of a parametrized curve c(t) in R3 , a ≤ t ≤ b: s=

Z b a

kc′ (t)k dt.

(1.2)

1.2 Representations of Inner Products by Symmetric Matrices

5

v u

θ

Fig. 1.1. The angle between two vectors.

Definition 1.1. An inner product on a real vector space V is a positive-definite, symmetric, bilinear form h , i : V × V → R. This means that for u, v, w ∈ V and a, b ∈ R, (i) (positive-definiteness) hv, vi ≥ 0; the equality holds if and only if v = 0. (ii) (symmetry) hu, vi = hv, ui. (iii) (bilinearity) hau + bv, wi = ahu, wi + bhv, wi. As stated, condition (iii) is linearity in only the first argument. However, by the symmetry property (ii), condition (iii) implies linearity in the second argument as well. Proposition 1.2 (Restriction of an inner product to a subspace). Let h , i be an inner product on a vector space V . If W is a subspace of V , then the restriction h , iW := h , i|W ×W : W × W → R is an inner product on W . ⊔ ⊓

Proof. Problem 1.3.

Proposition 1.3 (Nonnegative linear combination of inner products). Let h , ii , i = 1, . . . , r, be inner products on a vector V and let a1 , . . . , ar be nonnegative real numbers with at least one ai > 0. Then the linear combination h , i := ∑ ai h , ii is again an inner product on V . ⊔ ⊓

Proof. Problem 1.4.

1.2 Representations of Inner Products by Symmetric Matrices Let e1 , . . . , en be a basis for a vector space V . Relative to this basis we can represent vectors in V as column vectors:  1  1 y x  ..   ..  i i ∑ x ei ←→ x =  .  , ∑ y ei ←→ y =  .  . xn

yn

By bilinearity, an inner product on V is determined completely by its values on a set of basis vectors. Let A be the n × n matrix whose entries are

6


ai j = hei , e j i. By the symmetry of the inner product, A is a symmetric matrix. In terms of column vectors,

∑ xi ei , ∑ y j e j = ∑ ai j xi y j = xT Ay. Definition 1.4. An n × n symmetric matrix A is said to be positive-definite if

(i) xT Ax ≥ 0 for all x in Rn , and (ii) equality holds if and only if x = 0.

Thus, once a basis on V is chosen, an inner product on V determines a positivedefinite symmetric matrix. Conversely, if A is an n × n positive-definite symmetric matrix and {e1 , . . . , en } is a basis for V , then

∑ xi ei , ∑ yi ei = ∑ ai j xi y j = xT Ay defines an inner product on V . (Problem 1.1.) It follows that there is a one-to-one correspondence inner products on a vector n × n positive-definite ←→ . space V of dimension n symmetric matrices

The dual space V ∨ of a vector space V is by definition Hom(V, R), the space of all linear maps from V to R. Let α 1 , . . . , α n be the basis for V ∨ dual to the basis e1 , . . . , en for V . If x = ∑ xi ei ∈ V , then α i (x) = xi . Thus, with x = ∑ xi ei , y = ∑ y j e j , and hei , e j i = ai j , one has hx, yi = ∑ ai j xi y j = ∑ ai j α i (x)α j (y) = ∑ ai j (α i ⊗ α j )(x, y).

So in terms of the tensor product, an inner product h , i on V may be written as h , i = ∑ ai j α i ⊗ α j ,

where [ai j ] is an n × n positive-definite symmetric matrix.

1.3 Riemannian Metrics Definition 1.5. A Riemannian metric on a manifold M is the assignment to each point p in M of an inner product h , i p on the tangent space Tp M; moreover, the assignment p 7→ h , i p is required to be C∞ in the following sense: if X and Y are C∞ vector fields on M, then p 7→ hX p ,Yp i p is a C∞ function on M. A Riemannian manifold is a pair (M, h , i) consisting of a manifold M together with a Riemannian metric h , i on M. The length of a tangent vector v ∈ Tp M and the angle between two tangent vectors u, v ∈ Tp M on a Riemannian manifold are defined by the same formulas (1.1) and (1.2) as in R3 .

1.3 Riemannian Metrics

7

Example 1.6. Since all the tangent spaces Tp Rn for points p in Rn are canonically isomorphic to Rn , the Euclidean inner product on Rn gives rise to a Riemannian metric on Rn , called the Euclidean metric on Rn . Example 1.7. Recall that a submanifold M of a manifold N is said to be regular if locally it is defined by the vanishing of a set of coordinates [21, Section 9]. Thus, locally a regular submanifold looks like a k-plane in Rn . By a surface M in R3 we will mean a 2-dimensional regular submanifold of R3 . At each point p in M, the tangent space Tp M is a vector subspace of Tp R3 . The Euclidean metric on R3 restricts to a function h , iM : Tp M × Tp M → R, which is clearly positive-definite, symmetric, and bilinear. Thus a surface in R3 inherits a Riemannian metric from the Euclidean metric on R3 .

Recall that if F : N → M is a C∞ map of smooth manifolds and p ∈ N is a point in N, then the differential F∗ : Tp N → T f (p) M is the linear map of tangent spaces given by (F∗ X p )g = X p (g ◦ F) for any X p ∈ Tp N and any C∞ function g defined on a neighborhood of F(p) in M.

Definition 1.8. A C∞ map F : (N, h , i′ ) → (M, h , i) of Riemannian manifolds is said to be metric-preserving if for all p ∈ N and tangent vectors u, v ∈ Tp N, hu, vi′p = hF∗ u, F∗ viF(p) .

(1.3)

An isometry is a metric-preserving diffeomorphism. Example 1.9. If F : N → M is a diffeomorphism and h , i is a Riemannian metric on M, then (1.3) defines an induced Riemannian metric h , i′ on N. Example 1.10. Let N and M be the unit circle in C. Define F : N → M, a 2-sheeted covering space map, by F(z) = z2 . Give M a Riemannian metric h , i, for example, the Euclidean metric as a subspace of R2 , and define h , i′ on N by hv, wi′ = hF∗ v, F∗ wi. Then h , i′ is a Riemannian metric on N. The map F : N → M is metric-preserving but not an isometry because F is not a diffeomorphism. Example 1.11. A torus in R3 inherits the Euclidean metric from R3 . However, a torus is also the quotient space of R2 by the group Z2 acting as translations, or to put it more plainly, the quotient space of a square with the opposite edges identified (see [21, §7] for quotient spaces). In this way, it inherits a Riemannian metric from R2 . With these two Riemannian metrics, the torus becomes two distinct Riemannian manifolds (Figure 1.2). We will show later that there is no isometry between these two Riemannian manifolds with the same underlying torus.

8


Fig. 1.2. Two Riemannian metrics on the torus.

1.4 Existence of a Riemannian Metric A smooth manifold M is locally diffeomorphic to an open subset of a Euclidean space. The local diffeomorphism defines a Riemannian metric on a coordinate open set (U, x1 , . . . , xn ) by the same formula as for Rn . We will write ∂i for the coordinate vector field ∂ /∂ xi . If X = ∑ ai ∂i and Y = ∑ b j ∂ j , then the formula hX,Y i = ∑ ai bi

(1.4)

defines a Riemannian metric on U. To construct a Riemannian metric on M one needs to piece together the Riemannian metrics on the various coordinate open sets of an atlas. The standard tool for this is the partition of unity, whose definition we recall now. A collection {Sα } of subsets of a topological space S is said to be locally finite if every point p ∈ S has a neighborhood U p that intersects only finitely many of the subsets Sα . The support of a function f : S → R is the closure of the subset of S on which f 6= 0: supp f = cl{x ∈ S | f (x) 6= 0}. Suppose {Uα }α ∈A is an open cover of a manifold M. A collection of nonnegative C∞ functions ρα : M → R, α ∈ A, is called a C∞ partition of unity subordinate to {Uα } if

(i) supp ρα ⊂ Uα for all α , (ii) the collection of supports, {supp ρα }α ∈A , is locally finite, (iii) ∑α ∈A ρα = 1. The local finiteness of the supports guarantees that every point p has a neighborhood U p over which the sum in (iii) is a finite sum. (For the existence of a C∞ partition of unity, see [21, Appendix C].) Theorem 1.12. On every manifold M there is a Riemannian metric. Proof. Let {(Uα , φα )} be an atlas on M. Using the coordinates on Uα , we define as in (1.4) a Riemannian metric h , iα on Uα . Let {ρα } be a partition of unity subordinate to {Uα }. By the local finiteness of the collection {supp ρα }, every point p has a neighborhood U p on which only finitely many of the ρα ’s are nonzero.

1.4 Existence of a Riemannian Metric

9

Thus, ∑ ρα h , iα is a finite sum on U p . By Proposition 1.3, at each point p the sum ∑ ρα h , iα is an inner product on Tp M. To show that ∑ ρα h , iα is C∞ , let X and Y be C∞ vector fields on M. Since ∑ ρα hX,Y iα is a finite sum of C∞ functions on U p , it is C∞ on U p . Since p was ⊔ ⊓ arbitrary, ∑ ρα hX,Y iα is C∞ on M.

Problems 1.1.∗ Positive-definite symmetric matrix Show that if A is an n × n positive-definite symmetric matrix and {e1 , . . . , en } is a basis for V , then D E ∑ xi ei , ∑ yi ei = ∑ ai j xi y j = xT Ay

defines an inner product on V .

1.2.∗ Inner product Let V be an inner product space with inner product h , i. For u, v in V , prove that hu, wi = hv, wi for all w in V if and only if u = v. 1.3. Restriction of an inner product to a subspace Prove Proposition 1.2. 1.4.∗ Positive linear combination of inner products Prove Proposition 1.3. 1.5.∗ Extending a vector to a vector field Let M be a manifold. Show that for any tangent vector v ∈ Tp M, there is a C∞ vector field X on M such that X p = v. 1.6.∗ Equality of vector fields Suppose (M, h , i) is a Riemannian manifold. Show that two C∞ vector fields X,Y ∈ X(M) are equal if and only if hX, Zi = hY, Zi for all C∞ vector fields Z ∈ X(M). 1.7.∗ Upper half-plane Let H2 = {(x, y) ∈ R2 | y > 0}.

At each point p = (x, y) ∈ H2 , define

h , iH2 : Tp H2 × Tp H2 → R by 1 hu, vi, y2 where h , i is the usual Euclidean inner product. Show that h , iH2 is a Riemannian metric on H2 . hu, viH2 =

1.8. Product rule in Rn If f , g : R → Rn are differentiable vector-valued functions, show that h f , gi : R → R is differentiable and h f , gi′ = h f ′ , gi + h f , g′ i.

(Here f ′ means d f /dt.)

10


1.9. Product rule in an inner product space An inner product space (V, h , i) is automatically a normed vector space, with norm kvk = p hv, vi. The derivative of a function f : R → V is defined to be f ′ (t) = lim

h→0

f (t + h) − f (t) , h

provided that the limit exists, where the limit is taken with respect to the norm k k. If f , g : R → V are differentiable functions, show that h f , gi : R → R is differentiable and h f , gi′ = h f ′ , gi + h f , g′ i.

Appendices

§A Manifolds This appendix is a review, mostly without proofs, of the basic notions in the theory of manifolds and differential forms. For more details, see [21].

A.1 Manifolds and Smooth Maps We will be following the convention of classical differential geometry in which vector fields take on subscripts, differential forms take on superscripts, and coefficient functions can have either superscripts or subscripts depending on whether they are coefficient functions of vector fields or of differential forms. See [21, §4.7, p. 44] for a more detailed explanation of this convention. A manifold is a higher-dimensional analogue of a smooth curve or surface. Its prototype is the Euclidean space Rn , with coordinates r1 , . . . , rn . Let U be an open subset of Rn . A function f = ( f 1 , . . . , f m ) : U → Rm is smooth on U if the partial derivatives ∂ k f /∂ r j1 · · · ∂ r jk exist on U for all integers k ≥ 1 and all j1 , . . . , jk . In this book we use the terms “smooth” and “C∞ ” interchangeably. A topological space M is locally Euclidean of dimension n if for every point p in M, there is a homeomorphism φ of a neighborhood U of p with an open subset of Rn . Such a pair (U, φ : U → Rn ) is called a coordinate chart or simply a chart. If p ∈ U, then we say that (U, φ ) is a chart about p. A collection of charts {(Uα , φα : Uα → Rn )} is C∞ compatible if for every α and β , the transition function

φα ◦ φβ−1 : φβ (Uα ∩Uβ ) → φα (Uα ∩Uβ ) is C∞ . A collection of C∞ compatible charts {(Uα , φα : Uα → Rn )} that cover M is called a C∞ atlas. A C∞ atlas is said to be maximal if it contains every chart that is C∞ compatible with all the charts in the atlas. Definition A.1. A topological manifold is a Hausdorff, second countable, locally Euclidean topological space. By “second countable,” we mean that the space has

298

§A Manifolds

a countable basis of open sets. A smooth or C∞ manifold is a pair consisting of a topological manifold M and a maximal C∞ atlas {(Uα , φα )} on M. In this book all manifolds will be smooth manifolds. In the definition of a manifold, the Hausdorff condition excludes certain pathological examples, while the second countability condition guarantees the existence of a partition of unity, a useful technical tool that we will define shortly. In practice, to show that a Hausdorff, second countable topological space is a smooth manifold it suffices to exhibit a C∞ atlas, for by Zorn’s lemma every C∞ atlas is contained in a unique maximal atlas. Example A.2. Let S1 be the circle defined by x2 + y2 = 1 in R2 , with open sets (see Figure A.1) Ux+ = {(x, y) ∈ S1 | x > 0},

Ux− = {(x, y) ∈ S1 | x < 0},

Uy+ = {(x, y) ∈ S1 | y > 0},

Uy− = {(x, y) ∈ S1 | y < 0}.

Uy+ bc

bc

bc

Ux−

Ux+ bc

S1

Uy− Fig. A.1. A C∞ atlas on S1 .

Then {(Ux+ , y), (Ux− , y), (Uy+ , x), (Uy− , x)} is a C∞ atlas on S1 . For example, the transition function from the open interval ]0, 1[ = x(Ux+ ∩Uy− ) → y(Ux+ ∩Uy− ) = ] − 1, 0[

√ is y = − 1 − x2, which is C∞ on its domain.

A function f : M → Rn on a manifold M is said to be smooth or C∞ at p ∈ M if there is a chart (U, φ ) about p in the maximal atlas of M such that the function f ◦ φ −1 : Rm ⊃ φ (U) → Rn

A.2 Tangent Vectors

299

is C∞ . The function f : M → Rn is said to be smooth or C∞ on M if it is C∞ at every point of M. Recall that an algebra over R is a vector space together with a bilinear map µ : A × A → A, called multiplication, such that under addition and multiplication, A becomes a ring. Under addition, multiplication, and scalar multiplication, the set of all C∞ functions f : M → R is an algebra over R, denoted by C∞ (M). A map F : N → M between two manifolds is smooth or C∞ at p ∈ N if there is a chart (U, φ ) about p in N and a chart (V, ψ ) about F(p) in M with V ⊃ F(U) such that the composite map ψ ◦ F ◦ φ −1 : Rn ⊃ φ (U) → ψ (V ) ⊂ Rm is C∞ at φ (p). It is smooth on N if it is smooth at every point of N. A smooth map F : N → M is called a diffeomorphism if it has a smooth inverse, i.e., a smooth map G : M → N such that F ◦ G = 1M and G ◦ F = 1N . A typical matrix in linear algebra is usually an m × n matrix, with m rows and n columns. Such a matrix represents a linear transformation F : Rn → Rm . For this reason, we usually write a C∞ map as F : N → M, rather than F : M → N.

A.2 Tangent Vectors The derivatives of a function f at a point p in Rn depend only on the values of f in a small neighborhood of p. To make precise what is meant by a “small” neighborhood, we introduce the concept of the germ of a function. Decree two C∞ functions f : U → R and g : V → R defined on neighborhoods U and V of p to be equivalent if there is a neighborhood W of p contained in both U and V such that f agrees with g on W . The equivalence class of f : U → R is called the germ of f at p. It is easy to verify that addition, multiplication, and scalar multiplication are ∞ well-defined operations on the set C∞ p (M) of germs of C real-valued functions at p ∞ in M. These three operations make C p (M) into an algebra over R. Definition A.3. A point-derivation at a point p of a manifold M is a linear map ∞ D : C∞ p (M) → R such that for any f , g ∈ C p (M), D( f g) = (D f )g(p) + f (p)Dg. A point-derivation at p is also called a tangent vector at p. The set of all tangent vectors at p is a vector space Tp M, called the tangent space of M at p. Example A.4. If r1 , . . . , rn are the standard coordinates on Rn and p ∈ Rn , then the usual partial derivatives ∂ ∂ ,..., n ∂ r1 ∂r p

p

are tangent vectors at p that form a basis for the tangent space Tp (Rn ).

At a point p in a coordinate chart (U, φ ) = (U, x1 , . . . , xn ), where xi = ri ◦ φ is the ith component of φ , we define the coordinate vectors ∂ /∂ xi | p ∈ Tp M by ∂ ∂ f ◦ φ −1 f = i for each f ∈ C∞ p (M). ∂ xi p ∂ r φ (p)

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§A Manifolds

If F : N → M is a C∞ map, then at each point p ∈ N its differential F∗,p : Tp N → TF(p) M,

(A.1)

is the linear map defined by (F∗,p X p )(h) = X p (h ◦ F) ∞ (M). Usually the point p is clear from context and we for X p ∈ Tp N and h ∈ CF(p) may write F∗ instead of F∗,p . It is easy to verify that if F : N → M and G : M → P are C∞ maps, then for any p ∈ N,

(G ◦ F)∗,p = G∗,F(p) ◦ F∗,p , or, suppressing the points, (G ◦ F)∗ = G∗ ◦ F∗ .

A.3 Vector Fields A vector field X on a manifold M is the assignment of a tangent vector X p ∈ Tp M to each point p ∈ M. At every p in a chart (U, x1 , . . . , xn ), since the coordinate vectors ∂ /∂ xi | p form a basis of the tangent space Tp M, the vector X p can be written as a linear combination ∂ i X p = ∑ a (p) i with ai (p) ∈ R. ∂x p i

As p varies over U, the coefficients ai (p) become functions on U. The vector field X is said to be smooth or C∞ if M has a C∞ atlas such that on each chart (U, x1 , . . . , xn ) of the atlas, the coefficient functions ai in X = ∑ ai ∂ /∂ xi are C∞ . We denote the set of all C∞ vector fields on M by X(M). It is a vector space under the addition of vector fields and scalar multiplication by real numbers. As a matter of notation, we write tangent vectors at p as X p ,Yp , Z p ∈ Tp M, or if the point p is understood from context, as v1 , v2 , . . . , vk ∈ Tp M. As a shorthand, we sometimes write ∂i for ∂ /∂ xi . A frame of vector fields on an open set U ⊂ M is a collection of vector fields X1 , . . . , Xn on U such that at each point p ∈ U, the vectors (X1 ) p , . . . , (Xn ) p form a basis for the tangent space Tp M. For example, in a coordinate chart (U, x1 , . . . , xn ), the coordinate vector fields ∂ /∂ x1 , . . . , ∂ /∂ xn form a frame of vector fields on U. A C∞ vector field X on a manifold M gives rise to a linear operator on the vector space C∞ (M) of C∞ functions on M by the rule (X f )(p) = X p f

for f ∈ C∞ (M) and p ∈ M.

To show that X f is a C∞ function on M, it suffices to write X in terms of local coordinates x1 , . . . , xn in a neighborhood of p, say X = ∑ ai ∂ /∂ xi . Since X is assumed C∞ , all the coefficients ai are C∞ . Therefore, if f is C∞ , then X f = ∑ ai ∂ f /∂ xi is also.

A.4 Differential Forms

by

301

The Lie bracket of two vector fields X,Y ∈ X(M) is the vector field [X,Y ] defined [X,Y ] p f = X p (Y f ) − Yp (X f ) for p ∈ M and f ∈ C∞ p (M).

(A.2)

So defined, [X,Y ] p is a point-derivation at p (Problem A.3(a)) and therefore [X,Y ] is indeed a vector field on M. The formula for [X,Y ] in local coordinates (Problem A.3(b)) shows that if X and Y are C∞ , then so is [X,Y ]. If f : N → M is a C∞ map, its differential f∗,p : Tp N → T f (p) M pushes forward a tangent vector at a point in N to a tangent vector in M. It should be noted, however, that in general there is no push-forward map f∗ : X(N) → X(M) for vector fields. For example, when f is not one-to-one, say f (p) = f (q) for p 6= q in N, it may happen that for some X ∈ X(N), f∗,p X p 6= f∗,q Xq ; in this case, there is no way to define f∗ X so that ( f∗ X) f (p) = f∗,p X p for all p ∈ N. Similarly, if f : N → M is not onto, then there is no natural way to define f∗ X at a point of M not in the image of f . Of course, if f : N → M is a diffeomorphism, then the pushforward f∗ : X(N) → X(M) is well defined.

A.4 Differential Forms For k ≥ 1, a k-form or a form of degree k on M is the assignment to each p in M of an alternating k-linear function

ω p : Tp M × · · · × Tp M → R. {z } | k copies

Here “alternating” means that for every permutation σ of the set {1, 2, . . ., k} and v1 , . . . , vk ∈ Tp M,

ω p (vσ (1) , . . . , vσ (k) ) = (sgn σ )ω p (v1 , . . . , vk ),

(A.3)

where sgn σ , the sign of the permutation σ , is ±1 depending on whether σ is even or odd. We define a 0-form to be the assignment of a real number to each p ∈ M; in other words, a 0-form on M is simply a real-valued function on M. When k = 1, the condition of being alternating is vacuous. Thus, a 1-form on M is the assignment of a linear function ω p : Tp M → R to each p in M. For k < 0, a k-form is 0 by definition. An alternating k-linear function on a vector space V is also called a k-covector on V . As above, a 0-covector is a constant and a 1-covector on V is a linear function f : V → R. Let Ak (V ) be the vector space of all k-covectors on V . Then A0 (V ) = R and A1 (V ) = V ∨ := Hom(V, R), the dual vector space of V . In this language a k-form on M is the assignment of a k-covector ω p ∈ Ak (Tp M) to each point p in M. Let Sk be the group of all permutations of {1, 2, . . ., k}. A (k, ℓ)-shuffle is a permutation σ ∈ Sk+ℓ such that

σ (1) < · · · < σ (k) and σ (k + 1) < · · · < σ (k + ℓ). The wedge product of a k-covector α and an ℓ-covector β on a vector space V is by definition the (k + ℓ)-linear function

302

§A Manifolds

(α ∧ β )(v1 , . . . , vk+ℓ ) = ∑(sgn σ )α (vσ (1) , . . . , vσ (k) )β (vσ (k+1) , . . . , vσ (k+ℓ) ), (A.4) where the sum runs over all (k, ℓ)-shuffles. For example, if α and β are 1-covectors, then (α ∧ β )(v1 , v2 ) = α (v1 )β (v2 ) − α (v2 )β (v1 ). The wedge of a 0-covector, i.e., a constant c, with another covector ω is simply scalar multiplication. In this case, in keeping with the traditional notation for scalar multiplication, we often replace the wedge by a dot or even by nothing: c ∧ ω = c · ω = cω . The wedge product α ∧ β is a (k + ℓ)-covector; moreover, the wedge operation ∧ is bilinear, associative, and anticommutative in its two arguments. Anticommutativity means that α ∧ β = (−1)deg α deg β β ∧ α . Proposition A.5. If α 1 , . . . , α n is a basis for the 1-covectors on a vector space V , then a basis for the k-covectors on V is the set {α i1 ∧ · · · ∧ α ik | 1 ≤ i1 < · · · < ik ≤ n}. A k-tuple of integers I = (i1 , . . . , ik ) is called a multi-index. If i1 ≤ · · · ≤ ik , we call I an ascending multi-index, and if i1 < · · · < ik , we call I a strictly ascending multi-index. To simplify the notation, we will write α I = α i1 ∧ · · · ∧ α ik . As noted earlier, for a point p in a coordinate chart (U, x1 , . . . , xn ), a basis for the tangent space Tp M is ∂ ∂ ,..., n . ∂ x1 ∂x p

p

Let (dx1 ) p , . . . , (dxn ) p be the dual basis for the cotangent space A1 (Tp M) = Tp∗ M, i.e., ! ∂ i = δ ji . (dx ) p ∂ x j p

By Proposition A.5, if ω is a k-form on M, then at each p ∈ U, ω p is a linear combination: ω p = ∑ aI (p)(dxI ) p = ∑ aI (p)(dxi1 ) p ∧ · · · ∧ (dxik ) p . I

I

We say that the k-form ω is smooth if M has an atlas {(U, x1 , . . . , xn )} such that on each U, the coefficients aI : U → R of ω are smooth. By differential k-forms, we will mean smooth k-forms on a manifold. A frame of differential k-forms on an open set U ⊂ M is a collection of differential k-forms ω1 , . . . , ωr on U such that at each point p ∈ U, the k-covectors (ω1 ) p , . . . , (ωr ) p form a basis for the vector space Ak (Tp M) of k-covectors on the tangent space at p. For example, on a coordinate chart (U, x1 , . . . , xn ), the k-forms dxI = dxi1 ∧ · · · ∧ dxik , 1 ≤ i1 < · · · < ik ≤ n, constitute a frame of differential k-forms on U.

A.5 Exterior Differentiation on a Manifold

303

A subset B of a left R-module V is called a basis if every element of V can be written uniquely as a finite linear combination ∑ ri bi , where ri ∈ R and bi ∈ B. An R-module with a basis is said to be free, and if the basis is finite with n elements, then the free R-module is said to be of rank n. It can be shown that if a free R-module has a finite basis, then any two bases have the same number of elements, so that the rank is well defined. We denote the rank of a free R-module V by rkV . Let Ωk (M) denote the vector space of C∞ k-forms on M and let ∗

Ω (M) =

n M

Ωk (M).

k=0

If (U, x1 , .. . , xn ) is a coordinate chart on M, then Ωk (U) is a free module over C∞ (U) of rank nk , with basis dxI as above. L An algebra A is said to be graded if it can be written as a direct sum A = ∞ k=0 Ak of vector spaces such that under multiplication, Ak · Aℓ ⊂ Ak+ℓ . The wedge product ∧ makes Ω∗ (M) into an anticommutative graded algebra over R.

A.5 Exterior Differentiation on a Manifold An exterior derivative on a manifold M is a linear operator d : Ω∗ (M) → Ω∗ (M), satisfying the following three properties: (1) d is an antiderivation of degree 1, i.e., d increases the degree by 1 and for ω ∈ Ωk (M) and τ ∈ Ωℓ (M), d(ω ∧ τ ) = d ω ∧ τ + (−1)k ω ∧ d τ ; (2) d 2 = d ◦ d = 0; (3) on a 0-form f ∈ C∞ (M), (d f ) p (X p ) = X p f for p ∈ M and X p ∈ Tp M. By induction the antiderivation property (1) extends to more than two factors; for example, d(ω ∧ τ ∧ η ) = d ω ∧ τ ∧ η + (−1)deg ω ω ∧ d τ ∧ η + (−1)deg ω ∧τ ω ∧ τ ∧ d η . The existence and uniqueness of an exterior derivative on a general manifold is established in [21, Section 19]. To develop some facility with this operator, we will examine the case when M is covered by a single coordinate chart (U, x1 , . . . , xn ). This case can be used to define and compute locally on a manifold. Proposition A.6. Let (U, x1 , . . . , xn ) be a coordinate chart. Suppose d : Ω∗ (U) → Ω∗ (U) is an exterior derivative. Then (i) for any f ∈ Ω0 (U),

df =∑

∂f i dx ; ∂ xi

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§A Manifolds

(ii) d(dxI ) = 0; (iii) for any aI dxI ∈ Ωk (M), d(aI dxI ) = daI ∧ dxI . Proof. (i) Since (dx1 ) p , . . . , (dxn ) p is a basis of 1-covectors at each point p ∈ U, there are constants ai (p) such that (d f ) p = ∑ ai (p) (dxi ) p . Suppressing p, we may write d f = ∑ ai dxi . Applying both sides to the vector field ∂ /∂ xi gives ∂ ∂ i df = ∑ ai dx = ∑ ai δ ji = a j . j ∂xj ∂ x i i On the other hand, by property (3) of d, ∂ ∂ df = ( f ). j ∂x ∂xj Hence, a j = ∂ f /∂ x j and d f = ∑(∂ f /∂ x j ) dx j . (ii) By the antiderivation property of d, d(dxI ) = d(dxi1 ∧ · · · ∧ dxik ) = ∑(−1) j−1dxi1 ∧ · · · ∧ ddxi j ∧ · · · ∧ dxik j

=0

2

since d = 0.

(iii) By the antiderivation property of d, d aI dxI = daI ∧ dxI + aI d(dxI ) = daI ∧ dxI

since d(dxI ) = 0.

⊔ ⊓

Proposition A.6 proves the uniqueness of exterior differentiation on a coordinate chart (U, x1 , . . . , xn ). To prove its existence, we define d by two of the formulas of Proposition A.6: (i) if f ∈ Ω0 (U), then d f = ∑(∂ f /∂ xi ) dxi ; (iii) if ω = ∑ aI dxI ∈ Ωk (U) for k ≥ 1, then d ω = ∑ daI ∧ dxI . Next we check that so defined, d satisfies the three properties of exterior differentiation. (1) For ω ∈ Ωk (U) and τ ∈ Ωℓ (U), d(ω ∧ τ ) = (d ω ) ∧ τ + (−1)k ω ∧ d τ .

(A.5)

A.5 Exterior Differentiation on a Manifold

305

Proof. Suppose ω = ∑ aI dxI and τ = ∑ bJ dxJ . On functions, d( f g) = (d f )g + f (dg) is simply another manifestation of the ordinary product rule, since

∂ ( f g) dxi i ∂ x ∂f ∂g g + f i dxi =∑ ∂ xi ∂x ∂f ∂g = ∑ i dxi g + f ∑ i dxi ∂x ∂x = (d f ) g + f dg.

d( f g) = ∑

Next suppose k ≥ 1. Since d is linear and ∧ is bilinear over R, we may assume that ω = aI dxI and τ = bJ dxJ , each consisting of a single term. Then d(ω ∧ τ ) = d(aI bJ dxI ∧ dxJ )

= d(aI bJ ) ∧ dxI ∧ dxJ

(definition of d)

= (daI )bJ ∧ dxI ∧ dxJ + aI dbJ ∧ dxI ∧ dxJ (by the degree 0 case) = daI ∧ dxI ∧ bJ dxJ + (−1)k aI dxI ∧ dbJ ∧ dxJ = d ω ∧ τ + (−1)k ω ∧ d τ .

⊔ ⊓

(2) d 2 = 0 on Ωk (U). Proof. This is a consequence of the fact that the mixed partials of a function are equal. For f ∈ Ω0 (U), ! n n n ∂ f ∂2 f d 2 f = d ∑ i dxi = ∑ ∑ j i dx j ∧ dxi . i=1 ∂ x j=1 i=1 ∂ x ∂ x In this double sum, the factors ∂ 2 f /∂ x j ∂ xi are symmetric in i, j, while dx j ∧ dxi are skew-symmetric in i, j. Hence, for each pair i < j there are two terms

∂2 f dxi ∧ dx j , ∂ xi ∂ x j

∂2 f dx j ∧ dxi ∂ x j ∂ xi

that add up to zero. It follows that d 2 f = 0. For ω = ∑ aI dxI ∈ Ωk (U), where k ≥ 1, d 2 ω = d ∑ daI ∧ dxI

(by the definition of d ω )

= ∑(d 2 aI ) ∧ dxI + daI ∧ d(dxI ) = 0.

In this computation, d 2 aI = 0 by the degree 0 case, and d(dxI ) = 0 follows as in the ⊔ proof of Proposition A.6(ii) by the antiderivation property and the degree 0 case. ⊓

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§A Manifolds

(3) For f a C∞ function and X a C∞ vector field on (U, x1 , . . . , xn ), (d f )(X) = X f . Proof. Suppose X = ∑ a j ∂ /∂ x j . Then ∂f i ∂f j ∂ (d f )(X) = ∑ i dx = ∑ ai i = X f . a ∑ j ∂x ∂x ∂x

⊔ ⊓

A.6 Exterior Differentiation on R3 On R3 with coordinates x, y, z, every smooth vector field X is uniquely a linear combination ∂ ∂ ∂ X = a +b +c ∂x ∂y ∂z with coefficient functions a, b, c ∈ C∞ (R3 ). Thus, the vector space X(R3 ) of smooth vector fields on R3 is a free module of rank 3 over C∞ (R3 ) with basis {∂ /∂ x, ∂ /∂ y, ∂ /∂ z}. Similarly, Ω3 (R3 ) is a free module of rank 1 over C∞ (R3 ) with basis {dx ∧ dy ∧ dz}, while Ω1 (R3 ) and Ω2 (R3 ) are free modules of rank 3 over C∞ (R3 ) with bases {dx, dy, dz} and {dy ∧ dz, dz ∧ dx, dx ∧ dy} respectively. So the following identifications are possible: functions = 0-forms ←→ 3-forms f = f ←→ f dx ∧ dy ∧ dz and vector fields ↔ 1-forms ↔ 2-forms X = ha, b, ci ↔ a dx + b dy + c dz ↔ a dy ∧ dz + b dz ∧ dx + c dx ∧ dy. We will write fx = ∂ f /∂ x, fy = ∂ f /∂ y, and fz = ∂ f /∂ z. On functions, d f = fx dx + fy dy + fz dz. On 1-forms, d(a dx + b dy + c dz) = (cy − bz ) dy ∧ dz − (cx − az ) dz ∧ dx + (bx − ay ) dx ∧ dy. On 2-forms, d(a dy ∧ dz + b dz ∧ dx + c dx ∧ dy) = (ax + by + cz ) dx ∧ dy ∧ dz. Identifying forms with vector fields and functions, we have the following correspondences: d(0-form) ←→ gradient of a function,

d(1-form) ←→ curl of a vector field, d(2-form) ←→ divergence of a vector field.

A.7 Pullback of Differential Forms

307

A.7 Pullback of Differential Forms Unlike vector fields, which in general cannot be pushed forward under a C∞ map, differential forms can always be pulled back. Let F : N → M be a C∞ map. The pullback of a C∞ function f on M is the C∞ function F ∗ f := f ◦ F on N. This defines the pullback on C∞ 0-forms. For k > 0, the pullback of a k-form ω on M is the k-form F ∗ ω on N defined by (F ∗ ω ) p (v1 , . . . , vk ) = ωF(p) (F∗,p v1 , . . . , F∗,p vk ) for p ∈ N and v1 , . . . , vk ∈ Tp M. From this definition, it is not obvious that the pullback F ∗ ω of a C∞ form ω is C∞ . To show this, we first derive a few basic properties of the pullback. Proposition A.7. Let F : N → M be a C∞ map of manifolds. If ω and τ are k-forms and σ is an ℓ-form on M, then (i) F ∗ (ω + τ ) = F ∗ ω + F ∗ τ ; (ii) for any real number a, F ∗ (aω ) = aF ∗ ω ; (iii) F ∗ (ω ∧ σ ) = F ∗ ω ∧ F ∗ σ ; (iv) for any C∞ function h on M, dF ∗ h = F ∗ dh. Proof. The first three properties (i), (ii), (iii) follow directly from the definitions. To prove (iv), let p ∈ N and X p ∈ Tp N. Then (dF ∗ h) p (X p ) = X p (F ∗ h) = X p (h ◦ F)

(property (3) of d) (definition of F ∗ h)

and (F ∗ dh) p (X p ) = (dh)F(p) (F∗,p X p ) (definition of F ∗ ) = (F∗,p X p )h = X p (h ◦ F).

(property (3) of d) (definition of F∗,p)

Hence, dF ∗ h = F ∗ dh.

⊔ ⊓

We now prove that the pullback of a C∞ form is C∞ . On a coordinate chart (U, x1 , . . . , xn ) in M, a C∞ k-form ω can be written as a linear combination

ω = ∑ aI dxi1 ∧ · · · ∧ dxik , where the coefficients aI are C∞ functions on U. By the preceding proposition, F ∗ ω = ∑(F ∗ aI ) d(F ∗ xi1 ) ∧ · · · ∧ d(F ∗ xik )

= ∑(aI ◦ F) d(xi1 ◦ F) ∧ · · · ∧ d(xik ◦ F),

which shows that F ∗ ω is C∞ , because it is a sum of products of C∞ functions and C∞ 1-forms.

308

§A Manifolds

Proposition A.8. Suppose F : N → M is a smooth map. On C∞ k-forms, dF ∗ = F ∗ d. Proof. Let ω ∈ Ωk (M) and p ∈ M. Choose a chart (U, x1 , . . . , xn ) about p in M. On U, ω = ∑ aI dxi1 ∧ · · · ∧ dxik . As computed above, F ∗ ω = ∑(aI ◦ F) d(xi1 ◦ F) ∧ · · · ∧ d(xik ◦ F). Hence, dF ∗ ω = ∑ d(aI ◦ F) ∧ d(xi1 ◦ F) ∧ · · · ∧ d(xik ◦ F) = ∑ d(F ∗ aI ) ∧ d(F ∗ xi1 ) ∧ · · · ∧ d(F ∗ xik )

= ∑ F ∗ daI ∧ F ∗ dxi1 ∧ · · · ∧ F ∗ dxik

(dF ∗ = F ∗ d on functions by Prop. A.7(iv))

= ∑ F ∗ (daI ∧ dxi1 ∧ · · · ∧ dxik )

(F ∗ preserves the wedge product by Prop. A.7(iii)) = F ∗dω .

⊔ ⊓

In summary, for any C∞ map F : N → M, the pullback map F ∗ : Ω∗ (M) → Ω∗ (N) is an algebra homomorphism that commutes with the exterior derivative d. Example A.9 (Pullback under the inclusion map of an immersed submanifold). Let N and M be manifolds. A C∞ map f : N → M is called an immersion if for all p ∈ N, the differential f∗,p : Tp N → T f (p) M is injective. A subset S of M with a manifold structure such that the inclusion map i : S ֒→ M is an immersion is called an immersed submanifold of M. An example is the image of a line with irrational slope in the torus R2 /Z2 . An immersed submanifold need not have the subspace topology. If ω ∈ Ωk (M), p ∈ S, and v1 , . . . , vk ∈ Tp S, then by the definition of the pullback, (i∗ ω ) p (v1 , . . . , vk ) = ωi(p) (i∗ v1 , . . . , i∗ vk ) = ω p (v1 , . . . , vk ). Thus, the pullback of ω under the inclusion map i : S ֒→ M is simply the restriction of ω to the submanifold S. We also adopt the more suggestive notation ω |S for i∗ ω .

Problems A.1. Connected components (a) The connected component of a point p in a topological space S is the largest connected subset of S containing p. Show that the connected components of a manifold are open. (b) Let Q be the set of rational numbers considered as a subspace of the real line R. Show that the connected component of p ∈ Q is the singleton set {p}, which is not open in Q. Which condition in the definition of a manifold does Q violate?

A.7 Pullback of Differential Forms

309

A.2. Path-connectedness versus connectedness A topological space S is said to be locally path-connected at a point p ∈ S if for every neighborhood U of p, there is a path-connected neighborhood V of p such that V ⊂ U. The space S is locally path-connected if it is locally path-connected at every point p ∈ S. A path component of S is a maximal path-connected subset of S. (a) Prove that in a locally path-connected space S, every path component is open. (b) Prove that a locally path-connected space is path-connected if and only if it is connected. A.3. The Lie bracket Let X and Y be C∞ vector fields on a manifold M, and p a point in M. (a) Define [X,Y ] p by (A.2). Show that for f , g ∈ C∞ p (M), [X,Y ] p ( f g) = ([X,Y ] p f )g(p) + f (p)([X,Y ] p g). Thus, [X,Y ] p is a tangent vector at p. (b) Suppose X = ∑ ai ∂i and Y = ∑ b j ∂ j in a coordinate neighborhood (U, x1 , . . . , xn ) of p in M. Prove that [X,Y ] = ∑(a j ∂ j bi − b j ∂ j ai )∂i . i, j

Index

action, 247 action of GL(r, F) on polynomials, 312 Ad GL(r, R)-invariant, 216 adjoint representation of a Lie group, 125 adjoint bundle, 280 adjoint representation, 125, 252 of a Lie algebra, 126 affine connection, 45 algebra, 299 graded, 303 alternating k-linear function, 301 alternating linear map, 168 alternating multilinear maps universal mapping property, 170 angle between two vectors, 4 angle function, 142 anticommutativity, 302 antiderivation, 303 arc length, 4, 12, 131 is independent of parametrization, 131 arc length function, 12 arc length parametrization, 12 ascending multi-index, 302 associated bundle, 279 associativity of a bi-invariant metric, 127 of the tensor product, 165 atlas, 297 base space, 246 of a vector bundle, 51

basic iff invariant and horizontal, 284 basic form, 283 basis, 303 for a tensor product, 160 for the exterior power, 172 Betti numbers, 203 bi-invariant metric on a Lie group, 127 Bianchi identity first, 207 in vector form, 208 second, 208, 276 in vector form, 211 bilinear form, 5 bilinear maps universal mapping property, 157 bilinear maps over F, 61 binormal, 17 bundle isomorphism, 53 bundle map, 53 over a manifold, 53 ´ Cartan, Elie, 73 catenary, 38 catenoid, 39 characteristic classes, 216, 224 independence of a connection, 222 naturality of, 225 of a principal bundle, 294 vanishing, 227 characteristic form, 216 closed, 219

342

Index

chart, 297 chart about a point, 297 charts compatible, 297 Chern character, 319 Chern classes of a complex vector bundle, 239 of a principal GL(r, C)-bundle, 295 Chern–Weil homomorphism, 216, 218, 294 Christoffel symbols, 101, 102 for the Poincaré half-plane, 103 of a surface of revolution, 104 of the Poincaré disk, 104, 116 symmetric iff torsion-free, 102 circle volume form, 137 cobordant, 242 cocycle condition, 249 Codazzi–Mainardi equation, 64 coefficients of the first fundamental form, 37 of the second fundamental form, 38 coefficients of characteristic polynomial, 310 coefficients of characteristic polynomial are invariant polynomials, 313 coefficients of the characteristic polynomial, 216, 217 compatible charts, 297 compatible with the Hermitian metric, 238 complex inner product, 238 complex invariant polynomials generated by coefficients of characteristic polynials, 317 complex manifold, 241 complex vector bundle, 232 component, 33 congruent matrices, 234 connected component, 308 connection affine, 45 at a point, 79 compatible with the metric, 47, 77 defined using connection matrices, 213 Euclidean, 45 Levi-Civita, 47 metric in terms of forms, 84 on a complex vector bundle, 238

on a framed open set, 83 on a principal bundle, 258, 260 on a trivial bundle, 74 on a vector bundle, 74 Riemannian, 47 symmetric, 102 connection forms, 82 connection matrix, 82 dependence on frames, 206 connection on a vector bundle existence, 75 connection-preserving diffeomorphism, 101 preserves geodesics, 110 connections convex linear combination of, 50 convex linear combination, 75 coordinate chart, 297 coordinate vectors, 299 covariant derivative corresponding to a connection, 266 covariant derivative of tensor fields, 210 of a basic form, 292 of a tensorial form, 286 of a vector field along a curve, 99 on a principal bundle, 285 on surface in R3 , 106 covariant differentiation along a curve, 97 covector, 301 curvature, 46 G-equivariance, 275 and shape operator, 34 Gaussian, 20 Gaussian, in terms of an arbitrary basis, 66 geodesic curvature, 141 is F-linear, 46 is horizontal, 275 is independent of orientation, 15 mean, 20 normal, 20 of Maurer–Cartan connection, 278 of a connection on a principal bundle, 274 of a connection on a vector bundle, 76 of a plane curve, 14, 16 of a space curve, 17 of an ellipse, 16 of directional derivative, 26

Index principal, 20 sectional, 94 signed geodesic curvature, 143 symmetries, 209 total curvature, 149 total geodesic curvature, 144 curvature tensor, 211 independence of orthonormal basis, 94 curvature forms, 82 curvature matrix, 82 dependence on frames, 206 is skew-symmetric relative to an orthonormal frame, 86 curvature tensor, 76 skew-symmetry, 93 curve, 11 geometric, 11 parametrized, 11 piecewise smooth, 114 regular, 11 cuspidal cubic arc length, 17 cylinder mean and Gaussian curvature, 42 shape operator, 42 decomposable in the exterior algebra, 168 in the tensor product, 157 degree, 150 of a form, 192 of a line bundle, 244 diagonal entries of a skew-symmetric matrix is 0, 228 diagonalizable matrices are dense in gl(r, C), 316 diffeomorphism, 299 differential, 300 differential form, 301, 302 vector-valued, 191 differential forms depending smoothly on a real parameter, 220 with values in a vector bundle, 198 with values in a vector space, 190 differentiating under an integral sign), 222 dimension of tensor product, 161 direct sum

343

of vector bundles, 185 directional derivative computation using a curve, 24 in Rn , 24 has zero curvature, 27 is compatible with the metric, 27 is torsion-free, 27 of a vector field, 24 of a vector-valued form, 194 on a submanifold of Rn , 29 properties, 25 distance on a connected Riemannian manifold, 132 distribution, 258 horizontal, 255, 258 dot product, 4 dual of a module, 161 dual 1-forms under a change of frame, 135 dual 1-forms, 86 Ehresmann connection, 246, 260 Einstein summation convention, 82 elementary symmetric polynomials, 315 ellipse curvature, 16 equivariant map, 247 Euclid’s fifth postulate, 112 Euclid’s parallel postulate, 112 Euclidean connection, 45 Euclidean inner product, 4 Euclidean metric, 7 Euler characteristic independent of decomposition, 149 of a compact orientable odd-dimensional manifold is 0, 204 of a polygonal decomposition of a surface, 148 Euler class, 237 existence of a geodesic, 109 existence of a connection, 75 existence of a Hermitian metric, 238 existence of geodesics, 108 exponential map of a connection, 117 exponential map as a natural transformation, 129 differential, 119

344

Index

for a Lie group, 122, 124 naturality, 118 extension of algebraic identities, 311 exterior derivative, 303 exterior differentiation on R3 , 306 exterior algebra, 168 exterior derivative of a vector-valued 1-form, 202 of a vector-valued form, 194 properties, 304 exterior power, 168 basis, 172 F-bilinearity, 61 F-linear map of sections correspond to a bundle map, 60 fiber, 246 of a vector bundle, 51 fiber bundle, 246 first fundamental form coefficients, 37 first Bianchi identity, 207 vector form, 208 first fundamental form, 37 first fundamental form, 70 first structural equation, 87 flat section, 74 form smooth, 192, 302 forms with values in a Lie algebra, 195 frame, 58, 178 k-forms, 302 of vector fields, 300 positively oriented, 233 frame bundle, 251 of a vector bundle, 251 frame manifold of a vector space, 250 framed open set, 83 free action, 247 free module, 303 rank, 303 Frenet–Serret formulas, 17 Frenet–Serret frame, 17 functoriality of tensor product, 164

fundamental theorem on symmetric polynomials, 316 fundamental vector field, 251 integral curve, 253 right-equivariance, 252 vanishing at a point, 253 Fundamental vector fields Lie bracket of, 256 G-manifold, 247 Gauss curvature equation in terms of forms, 91 Gauss curvature equation, 64 Gauss map, 41, 42, 150 Gauss’s Theorema Egregium, 21, 65 Gauss–Bonnet formula for a polygon, 146 Gauss–Bonnet theorem, 22 for a surface, 148 generalized, 237 Gaussian curvature, 20, 69 and Gauss map, 150 in terms of an arbitrary basis, 66 is the determinant of the shape operator, 36 of a cylinder, 42 of a Riemannian 2-manifold, 93 of a sphere, 42 of a surface, 92 of a surface of revolution, 43 of the Poincaré disk, 116 Poincaré half-plane, 95 positive, 149 generalized Gauss–Bonnet theorem, 237 generalized second Bianchi identity, 208 on a frame bundle, 278 genus of a compact orientable surface, 149 geodesic, 97, 105 existence of, 108 existence of, 109 in the Poincaré half-plane, 110 maximal, 105 minimal, 133 on a sphere, 106 reparametrization, 107 speed is constant, 105 geodesic equations, 109 geodesic curvature, 141

Index signed geodesic curvature, 143 total geodesic curvature, 144 geodesic equations, 108 of the Poincaré disk, 116 geodesic polygon, 147 geodesic triangle sum of angles, 147 geodesically complete, 133 geodesics on a sphere, 109 geometric curve, 11 germ of a function at a point, 299 germ of neighborhoods, 129 graded algebra, 303 gradient vector field, 140 Gram–Schmidt process, 83 graph curvature, 16 helicoid, 39 Hermitian bundle, 238 Hermitian inner product, 238 Hermitian metric, 238 existence of, 238 Hermitian symmetric, 238 Hirzebruch–Riemann–Roch theorem, 244 holomorphic tangent bundle, 241 holomorphic vector bundle, 243 holonomy, 115 homogeneous elements, 167 homogeneous form, 192 homogeneous manifold, 248 Hopf bundle, 248 Hopf Umlaufsatz, 145 Hopf–Rinow theorem, 134 horizontal component of a form, 285 of a tangent vector, 260 horizontal distribution, 255, 258 of an Ehresmann connection, 261 horizontal form, 281 horizontal lift, 266, 267 of a vector field, 263, 270 horizontal lift formula, 270 horizontal lift of a vector field to a frame bundle, 270 to a principal bundle, 263 horizontal tangent vector, 268

345

horizontal vector, 266 Horizontal vector fields Lie bracket of, 278 hyperbolic plane, 116, 147 hyperbolic triangle, 147 hypersurface, 31, 68 normal vector field, 31 volume form, 140 immersed submanifold, 308 immersion, 308 induced connection on a pullback bundle, 214 inner product, 5 Euclidean, 4 representation by a symmetric matrix, 5 restriction to a subspace, 5 inner products nonnegative linear combination, 5 integral curve of a fundamental vector field, 253 integral form, 230 interior angle, 145 interior angles of a polygon, 150 invariant, 216 invariant complex polynomials, 314 invariant form, 284 invariant polynomial, 216, 310, 312 generators, 219 isometric invariant, 65 isometry, 3, 7 isomorphic vector bundles, 53 jump angle, 145 L-polynomials, 318 left action, 247 left G-equivariant map, 247 left-invariant metric, 126 left-invariant vector field, 121 Leibniz rule, 45 length, 6 of a vector, 4 length of a vector, 131 Levi-Civita connection, 47 Lie bracket of a vertical and a horizontal vector field, 264

346

Index

Lie bracket, 301, 309 of fundamental vector fields, 256 of horizontal vector fields, 278 of vector fields, 30 Lie derivative comparison with the directional derivative in Rn , 27 Lie group exponential map, 124 lift, 267 horizontal, 267 line bundle, 51, 239 trivial, 239 local operator, 55, 78 restriction, 57 local trivialization, 246 locally Euclidean, 297 locally finite, 8 locally path-connected, 309 locally trivial, 246 Maurer–Cartan connection, 264 curvature of, 278 Maurer–Cartan equation, 202 Maurer–Cartan form, 202 right translate, 202 maximal atlas, 297 maximal geodesic, 105 mean curvature, 20, 43, 69 of a cylinder, 42 of a sphere, 42 of a surface of revolution, 43 metric on the Poincaré half-plane, 95 metric connection, 77, 238 existence of, 78 relative to an orthonormal frame is skew-symmetric, 85 metric space, 132 metric-preserving map, 7 minimal geodesic, 133 Möbius strip, 52 module free, 303 morphism of principal bundles, 248 multi-index, 302 ascending, 302 strictly ascending, 302

natural transformation, 129 naturality of the exponential map, 118 of characteristic classes, 225 of the exponential map for a Lie group, 124 naturality property, 225 Newton’s identities, 321 Newton’s identity, 219 non-Euclidean geometry, 112 nondegenerate pairing, 173 normal coordinates, 120 normal curvature average value of, 43 of a normal section, 20 normal neighborhood, 120 normal section, 20 normal vector, 19 normal vector field, 19 along a hypersurface, 31 smooth, 19 orbit, 247 orientable vector bundle, 233 orientation and curvature, 15 on a vector bundle, 233 on a vector space, 232 orientation-preserving reparametrization, 131 orientation-reversing reparametrization, 131 oriented vector bundle, 233 orthogonal projection, 83 on a surface in R3 , 48 pairing nondegenerate, 173 of two modules, 173 parallel translation existence of, 113 parallel along a curve, 267 parallel frame along a curve, 267 parallel postulate, 112 parallel translate, 113 parallel translation, 113, 267 on a sphere, 115 preserves length and inner product, 114 parallel transport, 113, 267

Index parallel vector field, 112 parametrization by arc length, 12 parametrized curve, 11 partition of unity, 8 path component, 309 permutation matrix, 315 Pfaffian, 235 piecewise smooth curve, 114 Poincaré disk connection and curvature forms, 88 Poincaré half-plane Gaussian curvature, 95 metric, 95 Poincare Poincaré disk Christoffel symbols, 104 Poincaré disk Gaussian curvature, 96 Poincaré half-plane, 147 Poincaré half plane Gaussian curvature, 94 Poincaré half-plane geodesics, 110 point operator, 55 point-derivation, 299 polygon geodesic polygon, 147 on a surface, 145 polynomial, 216, 310 Ad(G)-invariant, 291 on gl(r, F), 312 on a vector space, 291 polynomial function, 310 polynomial on so(r), 234 polynomials versus polynomial functions, 310 Pontrjagin class, 229 Pontrjagin number, 230, 241 positive orientation on a polygon, 145 positive-definite, 238 positive-definite bilinear form, 5 positive-definite symmetric matrix, 6 positively oriented frame, 233 principal curvature is an eigenvalue of the shape operator, 35 principal bundle, 247 principal curvature, 20, 69

347

principal direction, 20 is an eigenvector of the shape operator, 35 principle of extension of algebraic identities, 311 product bundle, 52, 248 product of vector-valued forms, 192 projection orthogonal, 83 pseudo-tensorial form with respect to a representation, 281 pullback of a vector bundle, 181 of a differential form, 307 of a function, 307 pullback of vector-valued forms, 197 pullback bundle, 181 induced connection, 214 examples, 184 pushforward of of a vector field, 100 quotient bundle, 181 rank, 303 rank of a vector bundle, 51 real invariant polynomials, 319 generated by the coefficients of characteristic polynomial, 321 generation, 322 regular curve, 11 regular point, 31 regular submanifold, 7, 19 regular value, 31 reparametrization, 11, 131 orientation-preserving, 131 orientation-reversing, 131 reparametrization of a geodesic, 107 restriction of a connection to an open subset, 78 of a form to a submanifold, 308 of a local operator, 57 of a vector bundle to an open set, 182 of a vector bundle, 52 retraction, 256 Ricci curvature, 212 Riemann curvature tensor, 211 Riemannian bundle, 76 Riemannian connection, 47 existence and uniqueness, 47

348

Index

in terms of differential forms, 87 on a surface in R3 , 49 Riemannian manifold, 6 is a metric space, 132 Riemannian metric, 6 existence, 8 on a manifold with boundary, 136 on a vector bundle, 76 right action, 247 right G-equivariant map, 247 right-equivariant form with respect to a representation, 281 rotation angle theorem, 146 rotation index theorem, 146 scalar curvature, 213 second Bianchi identity in vector form, 211 second Bianchi identity, 208, 276 generalized, 208 generalized, on a frame bundle, 278 second fundamental form, 37, 70 coefficients, 38 second structural equation, 87 section of a vector bundle, 53 sectional curvature, 94 sections of a vector bundle along a curve, 266 sesquilinear, 238 shape operator, 32, 68 and curvature, 34 is self-adjoint, 33 matrix is symmetric, 34 of a cylinder, 42 of a sphere, 42 of a surface of revolution, 43 short exact sequence rank condition, 256 splitting, 256 shuffle, 176, 301 sign convention, 201 signature, 243 signed curvature, 14 signed geodesic curvature, 143 signs concerning vector-valued forms, 201 singular value, 31 skew-symmetric matrices powers, 228

skew-symmetric matrix diagonal 0, 228 smooth dependence of a form on t, 220 smooth form, 192, 302 smooth function, 297, 298 smooth manifold, 298 smooth map, 299 smooth vector field, 300 smoothly varying family of forms, 220 Ad SO(r) -invariant polynomial, 233 speed, 12, 105 sphere geodesics, 109 geodesics on, 106 mean and Gaussian curvature, 42 shape operator, 42 volume form in Cartesian coordinates, 138 spherical coordinates, 139 splitting, 255, 256 stabilizer, 247 straight, 97 strictly ascending multi-index, 302 structural equation first, 87 second, 82, 87 subbundle, 178 subbundle criterion, 179 submanifold immersed, 308 regular, 7 summation convention Einstein, 82 support of a function, 8 surface in R3 , 7 surface in R3 covariant derivative, 106 Gaussian curvature, 92 surface of revolution Christoffel symbols, 104 mean and Gaussian curvature, 43 shape operator, 43 symmetric bilinear form, 5 symmetric connection, 102 symmetric polynomials

Index fundamental theorem, 316 generation, 322 symmetric power, 177 symmetries of curvature, 209 tangent bundle, 53 tangent space, 299 tangent vector, 299 tensor, 93 tensor product basis, 160 tensor algebra, 166, 167 tensor criterior, 200 tensor field covariant derivative of, 210 tensor field on a manifold, 199 tensor product, 156 associativity, 165 basis, 160 characterization, 158 dimension, 161 functorial properties, 164 identities, 162 of finite cyclic groups, 163, 167 of three vector spaces, 165 tensorial form of type ρ , 281 Theorema Egregium, 21, 65 using forms, 92 third fundamental form, 70 topological manifold, 297 torsion, 46 is F-linear, 46 of directional derivative, 26 torsion forms, 86 torsion-free, 47 in terms of Christoffel symbols, 102 torsion-free connection, 102 torus as a Riemannian manifold, 7 total curvature, 42, 149 is a topological invariant, 149 of a plane curve, 150 total geodesic curvature, 144 total Pontrjagin class, 230 total space, 246 of a vector bundle, 51 trace

derivative of, 219 of a bilinear form, 213 trace polynomial, 217, 219, 310, 314 transition functions, 249 transitive action on a fiber, 248 transposition matrix, 315 triangle sum of angles, 147 trivial line bundle, 239 trivialization, 51, 74 trivializing open cover, 52 trivializing open set, 51 Umlaufsatz, 150 unit-speed polygon, 145 universal mapping property for bilinear maps, 158 universal mapping property for alternating k-linear maps, 170 for bilinear maps, 157 of the tensor product, 157 vanishing of characteristic classes, 227 vector bundle associated to a representation, 279 vector bundle, 51 base space, 51 holomorphic, 243 isomorphism, 53 total space, 51 vector field, 300 along a curve, 27 along a submanifold, 28 left-invariant, 121 on a submanifold, 28 parallel, 112 vector subbundle, 178 vector-valued forms product, 192 vector-valued k-covector, 190 vector-valued differential forms, 190 vector-valued form, 190, 191 directional derivative, 194 pullback, 197 velocity vector field, 27 vertical component, 258 of a tangent vector, 260 vertical subbundle, 255

349

350

Index

vertical tangent space, 254 vertical tangent vector, 254 volume form of a smooth hypersurface, 140 of a sphere, 138 of a sphere in spherical coordinates, 139 of an oriented Riemannian manifold, 136 on a circle, 137 on H2 , 136

on R2 , 136 on the boundary, 136 wedge product, 168, 301 is anticommutative, 169 properties, 168 under a change of frame, 135 wedge product formula, 176 Weingarten map, 32 Whitney product formula, 230

Loring W. Tu was born in Taipei, Taiwan, and grew up in Taiwan, Canada, and the United States. He attended McGill University and Princeton University as an undergraduate, and obtained his Ph.D. from Harvard University under the supervision of Phillip A. Griffiths. He has taught at the University of Michigan, Ann Arbor, and at Johns Hopkins University, and is currently Professor of Mathematics at Tufts University in Massachusetts. An algebraic geometer by training, he has done research at the interface of algebraic geometry, topolPhoto by Bachrach ogy, and differential geometry, including Hodge theory, degeneracy loci, moduli spaces of vector bundles, and equivariant cohomology. He is the coauthor with Raoul Bott of Differential Forms in Algebraic Topology and the author of An Introduction to Manifolds.

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