Graduate Texts in Mathematics S Axler 224 Editorial Board F.W Gehring K.A Ribet Springer Science+Business Media, LLC Graduate Texts in Mathematics 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 TAKEUTI/ZARING Introduction to Axiomatic Set Theory 2nd ed OXTOBY Measure and Category 2nd ed SCHAEFER Topological Vector Spaces 2nd ed HILTON/STAMMBACH A Course in Homological Algebra 2nd ed MAC LANE Categories for the Working Mathematician 2nd ed HUGHES/PIPER Projective Planes J.-P SERRE A Course in Arithmetic TAKEUTIIZARING Axiomatic Set Theory HUMPHREYS Introduction to Lie Algebras and Representation Theory COHEN A Course in Simple Homotopy Theory CONWAY Functions of One Complex Variable 2nd ed BEALS Advanced Mathematical Analysis ANDERSON/FuLLER Rings and Categories of Modules 2nd ed GOLUBITSKy/GUILLEMIN Stable Mappings and Their Singularities BERBERIAN Lectures in Functional Analysis and Operator Theory WINTER The Structure of Fields ROSENBLATT Random Processes 2nd ed HALMOS Measure Theory HALMOS A Hilbert Space Problem Book 2nd ed HUSEMOLLER Fibre Bundles 3rd ed HUMPHREYS Linear Algebraic Groups BARNES/MACK An Algebraic Introduction to Mathematical Logic GREUB Linear Algebra 4th ed HOLMES Geometric Functional Analysis and Its Applications HEWITT/STROMBERG Real and Abstract Analysis MANES Algebraic Theories KELLEY General Topology ZARISKl/SAMUEL Commutative Algebra Vol.! ZARISKl/SAMUEL Commutative Algebra Vol.lI JACOBSON Lectures in Abstract Algebra I Basic Concepts JACOBSON Lectures in Abstract Algebra II Linear Algebra JACOBSON Lectures in Abstract Algebra III Theory of Fields and Galois Theory HIRSCH Differential Topology 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 SPITZER Principles of Random Walk 2nd ed ALEXANDERIWERMER Several Complex Variables and Banach Algebras 3rd ed KELLEy/NAMIOKA et al Linear Topological Spaces MONK Mathematical Logic GRAUERT/FRITZSCHE Several Complex Variables ARVESON An Invitation to C*-Algebras KEMENy/SNELUKNAPP Denumerable Markov Chains 2nd ed ApOSTOL Modular Functions and Dirichlet Series in Number Theory 2nd ed J.-P SERRE Linear Representations of Finite Groups GILLMAN/JERI SON Rings of Continuous Functions KENDIG Elementary Algebraic Geometry LoiNE Probability Theory I 4th ed LoEVE Probability Theory II 4th ed MOISE Geometric Topology in Dimensions and SACHS/WU General Relativity for Mathematicians GRUENBERG/WEIR Linear Geometry 2nd ed EDWARDS Fermat's Last Theorem KLINGENBERG A Course in Differential Geometry HARTSHORNE Algebraic Geometry MANIN A Course in Mathematical Logic GRAVERIWATKINS Combinatorics with Emphasis on the Theory of Graphs BROWN/PEARCY Introduction to Operator Theory I: Elements of Functional Analysis MASSEY Algebraic Topology: An Introduction CROWELUFox Introduction to Knot Theory KOBLITZ p-adic Numbers, p-adic Analysis, and Zeta-Functions 2nd ed LANG Cyclotomic Fields ARNOLD Mathematical Methods in Classical Mechanics 2nd ed WHITEHEAD Elements of Homotopy Theory KARGAPOLOV/MERLZJAKOV Fundamentals of the Theory of Groups BOLLOBAS Graph Theory (continued after index) Gerard Walschap Metric Structures in Differential Geometry With 15 Figures Springer Gerard Walschap Department of Mathematics University of Oklahoma Norman, OK 73019-0315 USA gerard@math.ou.edu Editorial Board: S Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA axler@sfsu.edu F.W Gehring Mathematics Department East Hall University of Michigan Ann Arbor, MI 48109 USA fgehring@math.lsa.umich edu K.A Ribet Mathematics Department University of California, Berkeley Berkeley, CA 94720-3840 USA ribet@math.berkeley.edu Mathematics Subject Classification (2000): 53-xx, 58Axx 57Rxx Library of Congress Cataloging-in-Publication Data Walschap, Gerard 1954Metric structures in differential geometry/Gerard Walschap p cm Includes bibliographical references and index ISBN 978-1-4419-1913-7 ISBN 978-0-387-21826-7 (eBook) DOI 10.1007/978-0-387-21826-7 I Geometry, Differential l Title QA64I.W3272004 516.3'6-dc22 2003066219 Printed on acid-free paper © 2004 Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc in 2004 Softcover reprint of the hardcover I st edition 2004 All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights (EB) 54 I SPIN 10958674 Springer-Verlag is apart of Springer Science+Business Media springeronline.com Preface This text is an elementary introduction to differential geometry Although it was written for a graduate-level audience, the only requisite is a solid background in calculus, linear algebra, and basic point-set topology The first chapter covers the fundamentals of differentiable manifolds that are the bread and butter of differential geometry All the usual topics are covered, culminating in Stokes' theorem together with some applications The students' first contact with the subject can be overwhelming because of the wealth of abstract definitions involved, so examples have been stressed throughout One concept, for instance, that students often find confusing is the definition of tangent vectors They are first told that these are derivations on certain equivalence classes of functions, but later that the tangent space of ffi.n is "the same" as ffi n We have tried to keep these spaces separate and to carefully explain how a vector space E is canonically isomorphic to its tangent space at a point This subtle distinction becomes essential when later discussing the vertical bundle of a given vector bundle The following two chapters are devoted to fiber bundles and homotopy theory of fibrations Vector bundles have been emphasized, although principal bundles are also discussed in detail Special attention has been given to bundles over spheres because the sphere is the simplest base space for nontrivial bundles, and the latter can be explicitly classified The tangent bundle of the sphere, in particular, provides a clear and concrete illustration of the relation between the principal frame bundle and the associated vector bundle, and a short section has been specifically devoted to it Chapter studies bundles from the point of view of differential geometry, by introducing connections, holonomy, and curvature Here again, the emphasis is on vector bundles The last section discusses connections on principal bundles, and examines the relation between a connection on the frame bundle and that on the associated vector bundle Chapter introduces Euclidean bundles and Riemannian connections, and then embarks on a brief excursion into the realm of Riemannian geometry The basic tools, such as Levi-Civita connections, isometric immersions, Riemannian submersions, the Hopf-Rinow theorem, etc., are introduced, and should prepare the reader for more advanced texts on the subject The relation between curvature and topology is illustrated by the classical theorems of Hadamard-Cartan and Bonnet-Myers Chapter concludes with Chern-Weil theory, introducing the Pontrjagin, Euler, and Chern characteristic classes of a vector bundle In order to illustrate vi these concepts, vector bundles over spheres of dimension O f(p + tei) t - f(p) = (f c)'(O), where c is the line c(t) = P+tei through P in direction ei f is said to be smooth or differentiable on U if it has continuous partial derivatives of any order on U A map f : U -+ Rk is said to be smooth if all the component functions Ji := u i f : U -+ R of f are smooth In this case, the Jacobian matrix of f at P is the k x n matrix Df(p) whose (i,j)-th entry is DjJi(p) The Jacobian will often be identified with the linear transformation Rn -+ Rk it determines DEFINITION 1.1 A second countable Hausdorff topological space M is said to be a topological n-dimensional manifold if it is locally homeomorphic to Rn; i.e., if for any P E M there exists a homeomorphism x of some neighborhood U of P with some open set in Rn (U, x) is called a chart, or coordinate system, and 7: a coordinate map DEFINITION 1.2 A differentiable atlas on a topological n-dimensional manifold M is a collection A of charts of M such that (1) the domains of the charts cover M, and (2) if (U, x) and (V, y) E A, then y X~l : x(U n V) The map y (U, x) to (V, y) X~l -+ R n is smooth is often referred to as the transition map from the chart 214 CHARACTERISTIC CLASSES According to Theorem 8.1, 12k hand Pf oh are polynomials in These polynomials can be described explicitely: f1, , f~· PROPOSITION 8.6 12k oh = (_l)k L;~o( _l)lfU~k-l' Pf oh = (-1)ln/2]f~· PROOF By Lemma 8.1, n 1det(xln-M) 12 = det(h(x1n-M)) = det(xI2n -h(M)) = I>2(n-k) hkoh(M) k=O On the other hand, Idet(xln - M)I' ~ Idet(xln = = i( -iM))I' ~ I~( _1)'( _ilk !l Ep of Ep, we have that the matrix of R(x, y) in the basis (b, b) of Ep EB Ep is given by ( [Ra] a) [R] In the corresponding Hermitian basis (b, b) h, this matrix is just the original [R] In other words, if B : (Ep) > (n) is the isomorphism induced by b, and B : u(Ep EB Ep) > u(n) the one induced by (b, b) h, then BR = BR Thus, -i - 2k i - i k k2k g2k(R ) = f2k(BR) = hk(BR) = (-1) hk(BR) = (-1) g2dR ), which establishes the claim o To account for the odd Chern classes that are missing in the above theorem, define the conjugate bundle ~ of a complex bundle ~ to be the (complex) bundle with the same underlying total space and addition, but with scalar multiplication e given by aeu = au, where the right side is the usual scalar multiplication in~ Although the identity is a real bundle equivalence, ~ and its conjugate need not be equivalent as complex bundles; i.e., there may not be an equivalence h : ~ > ~ satisfying h(au) = a e h(u) = ah(u) Such an h does, however, exist when ~ is the complexification 7]1C of a real bundle 7]: It is straightforward to verify that the formula h( u, v) = (u, -v) defines such an equivalence PROPOSITION 9.1 If ( is a complex bundle, then the total Chern class of its conjugate is given by c(~) = - CI(O + C2(~) - C3(() + CHARACTERISTIC CLASSES 218 PROOF A Hermitian inner product (, I on ~ induces a Hermitian inner product (, I on ~ given by (U, VI := (U, VI = (V, UI, U, V E r~ A Hermitian connection 'V on ~ then becomes also a Hermitian connection 'V on the conjugate bundle, and their curvature tensors are related by (R(x, y)u, VI = (v, R(x, y)uI = (R(x, y)u, vI Since the eigenvalues of Rand R are imaginary, R(x,y) = -R(x,y) Thus, 9k(R) = 9k(-R) = (_1)k9k(R), o which establishes the claim We have seen that given a real bundle ~, its complexification ~iC is equivalent, in the complex sense, to the conjugate bundle ~iC' Proposition 9.1 then implies the following: COROLLARY 9.1 If ~ is a real bundle, then the odd Chern classes of its complexification are zero THEOREM PROOF 9.3 For complex bundles ~ and 71, c(~ EEl 71) = c(O U c(T/) Notice that for complex matrices M, N, k fk(A ® B) = L ft(A)fL[(B) [=0 The statement now follows by an argument similar to that in Theorem 4.2 EXAMPLE 9.2 Consider an oriented rank bundle ~o, and suppose its structure group reduces to S~ C SO(4); cf Section Thus, if P denotes the total space of the corresponding principal S3-bundle, then ~o = 7r : P XS3 lHl-+ + P / S~, with S~ acting on lHl = ]R4 by left multiplication Any quaternion q = a + bi + cj + dk can be written as (a + bi) + j(c - di) = Zl + jZ2 for some complex numbers Zl, Z2 The map Ii : lHl -+ ((:2, becomes a complex isomorphism if we define scalar multiplication in lHl by jZ2) := (Zl + jZ2)a = zla + jZ2a a(zl + The map Ii in turn induces a homomorphism Ii : GL(1,lHl) -+ GL(2,C) determined by Ii(qu) = li(q)li(u), q E lHl\ {O}, u E lHl + jZ2 E S3 and u = W1 + jW2 E lHl, qu = (Zl + jZ2)(W1 + jW2) = (ZlW1 - Z2W2) + j(Z2W1 + ZlW2) Recalling that ZlZl + Z2Z2 = 1, we conclude that Given q = Zl h(Zl + jZ2) = (Zl Z2 -Z2) _ E U(2) Zl CHERN CLASSES This exhibits ~o as the realification of a complex bundle Theorem 9.1, 219 ~ with group U(2) By PI (~o) = -(2C2(~) - CI (~) U cd~)) = 2e(~o) + ci(~)· Consider the map L : lHl -> lHl that sends q E lHl to qj L preserves addition, and given 0: E C, q = ZI + jZ2 E lHl, L(o:q) = L(ZlO: + jZ20:) = (ZIO: + jZ20:)j = zda + jZ2ja = (ZI + jZ2)ja =aLq Thus, L induces a complex equivalence ~ ~ ~, so that tion 9.1, and Pl(~O) = CI (0 = by Proposi- 2e(~o), a property already observed earlier in the special case that the base is a 4-sphere, cf Corollary 5.1 EXERCISE 164 Let ~ be a real vector bundle with complexification ~IC = J), J (u, v) = (-v, u) A Euclidean metric on ~ extends naturally to ~ EEl ~ by setting (~EEl ~, Ui , Vi E r~ By Exercise 161, there exists a unique Hermitian metric (,)c on ~IC the norm function of which equals that of the Euclidean metric Prove that + (U2' V2) + i( (U2, VI) Determine the total Chern class of 1'f,1 ((U1 , U2), (Vl' V2))c = (U1 , VI) EXERCISE 165 is just Cpl = ) - (UI, V2)) (observe that Cf,1 Bibliography [1] W Ballmann, Spaces of Nonpositive Curvature, Birkhiiuser, Basel 1995 [2] W Ballmann, V Schroeder, M Gromov, Manifolds of Nonpositive Curvature, Birkhiiuser, Boston 1985 [3] M Berger, B Gostiaux, Geometrie differentielle, Armand Colin, Paris 1972 [4] A Besse, Einstein Manifolds, Springer-Verlag, Berlin Heidelberg 1987 [5] R Bott, L W Tu, Differential Forms in Algebraic Topology, Springer-Verlag, New York 1982 [6] J.-P Bourguignon, H B Lawson, Jr., Stability and isolation phenomena for Yang-Mills theory, Comm Math Phys 79 (1982), 189-230 [7] J.-P Bourguignon, H B Lawson, Jr., J Simons, Stability and gap phenomena for YangMills fields, Proc Nat Acad Sci U.S.A 76 (1979), 1550-1553 [8] G E Bredon, Introduction to Compact Transformation Groups, Academic Press, New York 1972 [9] J Cheeger, Some examples of manifolds of nonnegative curvature, J Diff Geom (1973), 623-628 [10] J Cheeger, D Ebin, Comparison Theorems in Riemannian Geometry, North Holland, New York 1975 [11] J Cheeger, D Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann of Math 96 (1972), 413-443 [12] M Do Carmo, Differential Forms and Applications, Springer-Verlag, Berlin Heidelberg 1994 [13] _ _ _ , Riemannian Geometry, Birkhiiuser, Boston 1992 [14] S Gallot, D Hulin, J Lafontaine, Riemannian Geometry (2nd edition), Springer-Verlag, Berlin Heidelberg 1990 [15] D Gromoll, W Klingenberg, W Meyer, Riemannsche Geometrie im GrojJen (2nd edition), Springer-Verlag, Berlin Heidelberg 1975 [16] M Gromov, J Lafontaine, P Pansu, Structures metriques pour les varieUs Riemanniennes, Cedic/Fernand Nathan, Paris 1981 [17] S Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, New York 1962 [18] D Husemoller, Fiber Bundles (3rd edition), Springer-Verlag, New York 1994 [19] J Jost, Riemannian Geometry and Geometric Analysis, Springer, Berlin Heidelberg 1995 [20] S Kobayashi, Differential Geometry of Complex Vector Bundles, Iwanami Shoten and Princeton University Press, Princeton 1987 [21] S Kobayashi, K Nomizu, Foundations of Differential Geometry, John Wiley & Sons, New York 1963 [22] H B Lawson, Jr., The theory of Gauge fields in four dimensions, Amer Math Soc, CBMS 58, Providence 1985 [23] H B Lawson, Jr., M.-L Michelson, Spin Geometry, Princeton University Press, Princeton 1989 [24] J Milnor, Morse Theory, Princeton University Press, Princeton 1963 [25] _ _ _ , Topology from the Differentiable Viewpoint, University Press of Virginia, Charlottesville 1965 [26] J Milnor, J Stasheff, Characteristic Classes, Princeton University Press, Princeton 1974 221 222 BIBLIOGRAPHY [27] B O'Neill The fundamental equations of a submersion Michigan Math J 13 (1966), 459469 [28] _ _ _ , Semi-Riemannian Geometry Academic Press New York 1983 [29] 1\1 Ozaydm, G Walschap, Vector bundles with no soul, Proc Amer Math Soc 120 (1994), 565-567 [30] G Perelman, Proof of the soul conjecture of CheegeT and Gromoll J Differential Geom 40 (1994), 209-212 [31] P Petersen, Riemannian Geometry, Springer, New York 1998 [32] W A Poor, Differential Geometric StructuTes, McGraw-Hill, New York 1981 [33] A Rigas, Some bundles of nonnegative curvature, ~Iath Ann 232 (1978), 187-193 [34] ~! Spivak, A Comprehensive Introduction to Differential Geometry, Publish or Perish, Inc., Berkeley 1979 [35] N Steenrod, The Topology of Fiber Bundles, Princeton University Press, Princeton 1951 [36] F W \Varner, Foundations of Differentiable Manifolds and Lie Groups, Springer-Verlag, New York 1983 [37] G Whitehead, Elements of Homotopy Theory, Springer-Verlag, New York 1978 Index A-tensor, 150 classifying map, 77 space, 77, see also Grassmannian 79 complex structure, 69, 72, 208 ' complexification, 30, 208 of a real bundle, 215 conjugate point, 159 connection, 104 on dual bundle, ll3 Bott, 154 complete, 121 flat, 107, ll8 form, 128 Levi-Civita, 135 map, 109 on a manifold, 120 on principal bundle, 126 on Whitney sum, ll4 pullback, 105 Riemannian, ll4, 133 torsion of, 135 Yang-Mills, 198 coordinate system, covariant derivative, llO critical point, 14 cross section along a map, 105 zero, 68, 73, 81 cross-section, 67 cup product, 55 curvature form, 128 Gauss, 145 of bi-invariant metric, 144 operator, 142 principal, 145 Ricci,143 scalar, 143 sectional, 143 tensor, 114 anti-self-dual, 192 of bi-invariant metric, 139 adjoint representation, 22 algebra exterior, 32 of contravariant tensors, 32 tensor, 32 alternating map, 33 Bianchi identity, 120, 129 Bonnet-Myers, 172 boundary operator, 84, 92 bundle associated, 63 complex, 69 conjugate, 217 coordinate, 57 cotangent, 10 dual,67 equivalence, 59, 65, 66, 70 Euclidean, 131 exterior algebra, 35 fiber, 59 frame, 63, 65, 129 homomorphism, 68, 70 map, 59, 109 normal, 72, 73, 81 of tensors, 35 principal, 60, 62 pullback, 69, 72, 76 stably trivial, 191 sub, 64 tangent, 10, 58, 66, 76 of homogeneous space, 66 tensor product, 68 unit sphere, 80 universal, 76, 79 vertical, 72, 104, 199 Cartan's structure equation, ll6, 128 chart, Chern class, 215 Christoffel symbols, 120 223 INDEX 224 of product metric, 140 self-dual, 192 decomposable elements, 31 degree, 54, 55, 86, 96, 97, 99, 191, 203 derivative, 8, 11 exterior, 37, 40, 115 exterior covariant, 115 diffeomorphism, differentiable atlas, bundle, 57 differentiable map, degree of, 54, 86 equivariant, 60 differentiable structure, differential, 9, 37 differential form, 35, 67 anti-self-dual, 188, 192 bundle-valued, 113 closed, 38, 44, 53, 55 exact, 38 self-dual, 188, 192 distribution, 27 horizontal, 149 integrable, 27 vertical, 149 divergence, 139 Euler characteristic, 203 Euler class, 187 Euler number, 191 exact sequence, 71 exponential map for matrices, 124 of a spray, 122 of a Lie group, 125 fibration, 73, 89 flow, 18, 23, 25 foliation, 28 Frobenius theorem, 28 Gauss equations, 145 geodesic, 120 minimal, 163 gradient, 140 graph, 16 Grassmannian, 75 complex, 80 homotopy groups of, 90 oriented, 80 Hadamard-Cartan, 172 Hermitian inner product, 208 Hessian form, 140 tensor, 140 Hodge star operator, 188 holonomy group, 106 homogeneous space, 60, 67, 75 invariant metric on, 132 normal, 152 homotopy, 51, 73 covering property, 75 group, 84 lifting property, 73, 74 sequence, 85, 89 Hopf fibration, 29, 58, 60, 67, 80, 97 Euler and Pontrjagin numbers of, 197 Hopf-Rinow theorem, 164 horizontal lift, 149 imbedding, 12 immersion, 12 isometric, 137 implicit function theorem, 11 index form, 168 injectivity radius, 163 integral over a chain, 42 over an oriented manifold, 48 integral curve, 17 inverse function theorem, 11 isometry, 131 isotropy group, 66, 173 isotropy representation, 66 Jacobi identity, 20 Jacobian matrix, Killing form, 206 Laplacian, 140 leaf, 28 Lie algebra, 20, 21, 26, 34 Lie bracket, 20 Lie derivative, 23, 40 of tensor field, 141 Lie group, 21, 50 action, 46 effective, 57, 66 free, 47, 60 proper, 176 properly discontinuous, 47 integration on, 50 semi-simple, 206 manifold product manifold, topological, differentiable, Einstein, 205 integral, 27, 107 parallelizable, 103, 121 Riemannian, 131 Stiefel, 79 INDEX homotopy groups of, 93 with boundary, 48 metric bi-invariant, 131, 207 Einstein, 205 Euclidean, 68 on dual, 133 on exterior algebra bundle, 133 on homomorphism bundle, 133 Hermitian, 215 left-invariant, 131 Riemannian, 36, 68 conformally equivalent, 144 product, 132 Whitney sum, 132 multilinear map, 30 nonsingular pairing, 31 one-parameter group of diffeomorphims, 19 orbit, 46 orient able bundle, 65, 90 manifold, 46 orientation, 45 orthogonal group, 26 stable homotopy group of, 90 parallel translation, 103, 106 partition of unity, 3, 48 Pfaffian, 184 Poincare Lemma, 51, 178 polynomial, 179 invariant, 179 polarization of, 179 Pontrjagin class, 183 Pontrjagin number, 191 product tensor, 29 wedge, 32 projective space complex, 59 complex curvature of, 153 quaternionic, 59 real, 47, 67, 75 quaternions, 22, 59, 66 realification, 207 of a complex bundle, 215 regular point, 14 regular value, 14, 15 Ricci curvature, 143 Sard's theorem, 15 scalar curvature, 143 225 second fundamental tensor, 145 sectional curvature, 143 of bi-invariant metric, 144 of sphere, 144, 147 singular chain, 42 boundary of, 42 cube, 41 sphere, 2, 46, 55 bundles over characteristic class of, 94, 95 characteristic map of, 94, 99 canonical connection on, 107, 113 curvature tensor of, 119, 144 Euler class of, 189 geodesics on, 122 homotopy groups of, 86, 90 Pontrjagin class of, 183 Spin, 69, 96, 101 spin structure, 96, 101 spray, 122 geodesic, 124 structure group, 57 reduction of, 65, 68 submanifold, 13 canonical connection on, 113 immersed, 13 Riemannian, 138 totally geodesic, 147 submersion, 15, 47, 60, 147 homogeneous, 152 Riemannian, 149 symmetric function, 178 elementary, 178 symplectic form, 209 tangent space, tangent vector, tensor, 31 tensor field, 35 transition function, 57 transverse regular, 16 tubular neighborhood, 81, 174 unitary group, 80, 211 variation piecewise-smooth, 167 vector bundle, 58, see also bundle complex, 215 vector field, 16, 67, see also cross-section basic, 149 complete, 19 coordinate vector fields, 6, 16 divergence of, 139 index of, 203 Jacobi,156 226 Killing, 141, 160 left-invariant, 21, 23, 125 position, 23, 46 volume form, 46, 51, 139 Whitney sum, 70 Euler class of, 190 metric, 132 Pontrjagin class of, 190 INDEX Graduate Texts in Mathematics (continuedfrom page ii) 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 EDWARDS Fourier Series Vol I 2nd ed WELLS Differential Analysis on Complex Manifolds 2nd ed WATERHOUSE Introduction to Affine Group Schemes SERRE Local Fields WEIDMANN Linear Operators in Hilbert Spaces LANG Cyclotomic Fields II MASSEY Singular Homology Theory FARKAs/KRA Riemann Surfaces 2nd ed STILLWELL Classical Topology and Combinatorial Group Theory 2nd ed HUNGERFORD Algebra DAVENPORT Multiplicative Number Theory 3rd ed HOCHSCHILD Basic Theory of Algebraic Groups and Lie Algebras IITAKA Algebraic Geometry HECKE Lectures on the Theory of Algebraic Numbers BURRIS/SANKAPPANAVAR A Course in Universal Algebra WALTERS An Introduction to Ergodic Theory ROBINSON A Course in the Theory of Groups 2nd ed FORSTER Lectures on Riemann Surfaces BOTT/TU Differential Forms in Algebraic Topology WASHINGTON Introduction to Cyclotomic Fields 2nd ed IRELAND/ROSEN A Classical Introduction to Modern Number Theory 2nd ed EDWARDS Fourier Series Vol II 2nd ed VAN LINT Introduction to Coding Theory 2nd ed BROWN Cohomology of Groups PIERCE Associative Algebras LANG Introduction to Algebraic and Abelian Functions 2nd ed BR0NDSTED An Introduction to Convex Polytopes BEARDON On the Geometry of Discrete Groups DIESTEL Sequences and Series in Banach Spaces DUBROVIN/FoMENKOINOVIKOV Modern Geometry-Methods and Applications Part I 2nd ed WARNER Foundations of Differentiable Manifolds and Lie Groups SHIRYAEV Probability 2nd ed 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 CONWAY A Course in Functional Analysis 2nd ed KOBLITZ Introduction to Elliptic Curves and Modular Forms 2nd ed BROCKERIToM DIECK Representations of Compact Lie Groups GRovE/BENSON Finite Reflection Groups 2nd ed BERG/CHRISTENSEN/REsSEL Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions EDWARDS Galois Theory VARADARAJAN Lie Groups, Lie Algebras and Their Representations LANG Complex Analysis 3rd ed DUBROVINlFoMENKOlNovIKOV Modem Geometry-Methods and Applications Part II LANG SL2(R) SILVERMAN The Arithmetic of Elliptic Curves OLVER Applications of Lie Groups to Differential Equations 2nd ed RANGE Holomorphic Functions and Integral Representations in Several Complex Variables LEHTO Univalent Functions and Teichmilller Spaces LANG Algebraic Number Theory HUSEMOLLER Elliptic Curves 2nd ed LANG Elliptic Functions KARATZAS/SHREVE Brownian Motion and Stochastic Calculus 2nd ed KOBLITZ A Course in Number Theory and Cryptography 2nd ed BERGERIGOSTIAUX Differential Geometry: Manifolds, Curves, and Surfaces KELLEy/SRINIVASAN Measure and Integral Vol I J.-P SERRE Algebraic Groups and Class Fields PEDERSEN Analysis Now ROTMAN An Introduction to Algebraic Topology ZIEMER Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation LANG Cyclotomic Fields I and II Combined 2nd ed REMMERT Theory of Complex Functions Readings in Mathematics EBBINGHAuslHEJlMES et al Numbers Readings in Mathematics 124 DUBROVIN/FoMENKO!NovIKOV Modem Geometry-Methods and Applications Part III 125 BERENSTEIN/GAY Complex Variables: An Introduction 126 BOREL Linear Algebraic Groups 2nd ed 127 MASSEY A Basic Course in Algebraic Topology 128 RAUCH Partial Differential Equations 129 FULTON/HARRIS Representation Theory: A First Course Readings in Mathematics 130 DODSON/POSTON Tensor Geometry 131 LAM A First Course in Noncommutative Rings 2nd ed 132 BEARDON Iteration of Rational Functions 133 HARRIS Algebraic Geometry: A First Course 134 ROMAN Coding and Information Theory 135 ROMAN Advanced Linear Algebra 136 ADKINS/WEINTRAUB Algebra: An Approach via Module Theory 137 AXLERIBoURDON/RAMEY Harmonic Function Theory 2nd ed 138 COHEN A Course in Computational Algebraic Number Theory 139 BREDON Topology and Geometry 140 AUBIN Optima and Equilibria An Introduction to Nonlinear Analysis 141 BECKERIWEISPFENNING/KREDEL Grabner Bases A Computational Approach to Commutative Algebra 142 LANG Real and Functional Analysis 3rd ed 143 DOOB Measure Theory 144 DENNIS/F ARB Noncommutative Algebra 145 VICK Homology Theory An Introduction to Algebraic Topology 2nd ed 146 BRIDGES Computability: A Mathematical Sketchbook 147 ROSENBERG Algebraic K-Theory and Its Applications 148 ROTMAN An Introduction to the Theory of Groups 4th ed 149 RATCLIFFE Foundations of Hyperbolic Manifolds ISO EISENBUD Commutative Algebra with a View Toward Algebraic Geometry lSI SILVERMAN Advanced Topics in the Arithmetic of Elliptic Curves 152 ZIEGLER Lectures on Polytopes 153 FULTON Algebraic Topology: A First Course 154 BROWN/PEARCY An Introduction to Analysis ISS KASSEL Quantum Groups 156 KECHRIS Classical Descriptive Set Theory 157 MALLIAVIN Integration and Probability 158 ROMAN Field Theory 159 CONWAY Functions of One Complex Variable II 160 LANG Differential and Riemannian Manifolds 161 BORWEIN/ERDELYI Polynomials and Polynomial Inequalities 162 ALPERIN/BELL Groups and Representations 163 DIXON/MoRTIMER Permutation Groups 164 NATHANSON Additive Number Theory: The Classical Bases 165 NATHANSON Additive Number Theory: Inverse Problems and the Geometry of Sumsets 166 SHARPE Differential Geometry: Cartan's Generalization of Klein's Erlangen Program 167 MORANDI Field and Galois Theory 168 EWALD Combinatorial Convexity and Algebraic Geometry 169 BHATIA Matrix Analysis 170 BREDON Sheaf Theory 2nd ed 171 PETERSEN Riemannian Geometry 172 REMMERT Classical Topics in Complex Function Theory 173 DIESTEL Graph Theory 2nd ed 174 BRIDGES Foundations of Real and Abstract Analysis 175 LICKORISH An Introduction to Knot Theory 176 LEE Riemannian Manifolds 177 NEWMAN Analytic Number Theory 178 CLARKE/LEDY AEV/STERN/WOLENSKI Nonsmooth Analysis and Control Theory 179 DOUGLAS Banach Algebra Techniques in Operator Theory 2nd ed 180 SRIVASTAVA A Course on Borel Sets 181 KRESS Numerical Analysis 182 WALTER Ordinary Differential Equations 183 MEGGINSON An Introduction to Banach Space Theory 184 BOLLOBAS Modern Graph Theory 185 COx/LITTLE/O'SHEA Using Algebraic Geometry 186 RAMAKRISHNANN ALENZA Fourier Analysis on Number Fields 187 HARRIs/MORRISON Moduli of Curves 188 GOLDBLATT Lectures on the Hyperreals: An Introduction to Nonstandard Analysis 189 LAM Lectures on Modules and Rings 190 ESMONDE/MuRTY Problems in Algebraic Number Theory 191 LANG Fundamentals of Differential Geometry 192 HIRSCH/LACOMBE Elements of Functional Analysis 193 COHEN Advanced Topics in Computational Number Theory 194 ENGELINAGEL One-Parameter Semi groups for Linear Evolution Equations 195 NATHANSON Elementary Methods in Number Theory 196 OSBORNE Basic Homological Algebra 197 EISENBUD/HARRIs The Geometry of Schemes 198 ROBERT A Course in p-adic Analysis 199 HEDENMALMIKORENBWMlZHU Theory of Bergman Spaces 200 BAO/CHERN/SHEN An Introduction to Riemann-Finsler Geometry 201 HINDRY/SILVERMAN Diophantine Geometry: An Introduction 202 LEE Introduction to Topological Manifolds 203 SAGAN The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions 204 ESCOFIER Galois Theory 205 FELIx/HALPERIN/THOMAS Rational Homotopy Theory 2nd ed 206 MURTY Problems in Analytic Number Theory Readings in Mathematics 207 GODSILIROYLE Algebraic Graph Theory 208 CHENEY Analysis for Applied Mathematics 209 ARVESON A Short Course on Spectral Theory 210 ROSEN Number Theory in Function Fields 211 LANG Algebra Revised 3rd ed 212 MATOUSEK Lectures on Discrete Geometry 213 FRlTZSCHE/GRAUERT From Holomorphic Functions to Complex Manifolds 214 lOST Partial Differential Equations 215 GOLDSCHMIDT Algebraic Functions and Projective Curves 216 D SERRE Matrices: Theory and Applications 217 MARKER Model Theory: An Introduction 218 LEE Introduction to Smooth Manifolds 219 MACLACHLAN/REID The Arithmetic of Hyperbolic 3-Manifolds 220 NESTRUEV Smooth Manifolds and Observables 221 GRONBAUM Convex Polytopes 2nd ed 222 HALL Lie Groups, Lie Algebras, and Representations: An Elementary Introduction 223 VRETBLAD Fourier Analysis and Its Applications 224 WALSCHAP Metric Structures in Differential Geometry ... Library of Congress Cataloging -in- Publication Data Walschap, Gerard 195 4Metric structures in differential geometry /Gerard Walschap p cm Includes bibliographical references and index ISBN 978-1-4419-1913-7... GRUENBERG/WEIR Linear Geometry 2nd ed EDWARDS Fermat's Last Theorem KLINGENBERG A Course in Differential Geometry HARTSHORNE Algebraic Geometry MANIN A Course in Mathematical Logic GRAVERIWATKINS Combinatorics... proprietary rights (EB) 54 I SPIN 10958674 Springer-Verlag is apart of Springer Science+Business Media springeronline.com Preface This text is an elementary introduction to differential geometry Although