Differential and riemannian manifolds, serge lang

Graduate Texts in Mathematics 160 Editorial Board S Axler F.W Gehring P.R Halmos Springer New York Berlin Heidelberg Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo BOOKS OF RELATED INTEREST BY SERGE LANG Linear Algebra, Third Edition 1987, ISBN 96412-6 Undergraduate Algebra, Second Edition 1990, ISBN 97279-X Complex Analysis, Third Edition 1993, ISBN 97886-0 Real and Functional Analysis, Third Edition 1993, ISBN 94001-4 Algebraic Number Theory, Second Edition 1994, ISBN 94225-4 Introduction to Complex Hyperbolic Spaces 1987, ISBN 96447-9 OTHER BOOKS BY LANG PUBLISHED BY SPRINGER-VERLAG Introduction to Arakelov Theory • Riemann-Roch Algebra (with William Fulton) • Complex Multiplication • Introduction to Modular Forms • Modular Units (with Daniel Kubert) • Fundamentals of Diophantine Geometry • Elliptic Functions • Number Theory III • Cyclotomic Fields I and II • SL (R) • Abelian Varieties • Introduction to Algebraic and Abelian Functions • Undergraduate Analysis • Elliptic Curves: Diophantine Analysis • Introduction to Linear Algebra • Calculus of Several Variables • First Course in Calculus • Basic Mathematics • Geometry: A High School Course (with Gene Murrow) • Math! Encounters with High School Students • The Beauty of Doing Mathematics • THE FILE Serge Lang Differen tial and RieInannian Manifolds With 20 Illustrations Springer Serge Lang Department of Mathe matics Yale University New Haven, CT 06520 USA Editorial Board S Ax ler Depa.rtment o f Mathematics Michigan State Unive rsit y East Lansing, MI 48824 USA F.W Geh rin g Department of Mathematics U ni versity o f Michigan Ann Arbor, MI 48109 USA P.R Halmos Department of Mathematics Santa Clara University Santa Clara, CA 95053 USA Mathematics Subject Classifications Code: 58'()1 Library of Congress Cataloging-in-Publkation Data Lang, Serge 1927Differential and Riemannian manifolds I Serge Lang p cm - (G raduate texts in mathematics; 160) Includes bibliographical references (p ) and index ISBN- 13: 978- 1-46 12-8688-2 I Differentiable manifolds Riemannian manifo lds I Title II Series QA614.3.L34 1995b 5]6 3'6- IAI, then lAm < and hence series argument) So is A - ~ and is in the general spectrum Then ~ general spectrum is open, because if to ~o, then (A - ~orl(A - ~) is close - ~ is also invertible Furthermore, if I - (A/~) is invertible (by the power we are done Finally, suppose that ~ is real Otherwise, let g(t) = (t - ~)(t - ~) Then g(t) =I- on O"(A) and h(t) = l/g(t) is its inverse From this we see that A - ~ is invertible Suppose ~ is not in the spectrum Then t - ~ is invertible and so is A -~ Suppose ~ is in the spectrum After a translation, we may suppose that is in the spectrum Consider the function g(t) as follows: g (t) = {lfltl, N, It I ~ liN, It I ~ liN, (g is positive and has a peak at 0.) If A is invertible, BA = I, then from Itg(t)1 ~ we get IAg(A)1 ~ and hence Ig(A)1 ~ IBI But g(A) becomes arbitrarily large as we take N large Contradiction Theorem 3.11 Let S be a set of operators of the Hilbert space E, leaving no closed subspace invariant except and E itself Let A be a Hermitian operator such that AB = BA for all BE S Then A = AI for some real number A [APP., §3] HERMITIAN OPERA TORS 353 Proof It will suffice to prove that there is only one element in the spectrum of A Suppose there are two, iiI #- 1i • There exist continuous functions f, g on the spectrum such that neither is on the spectrum, but fg is on the spectrum For instance, one may take for f, g the functions whose graph is indicated on the next diagram f We have f(A)B = Bf(A) for all BE S (because B commutes with real polynomials in A, hence with their limits) Hence f(A)E is invariant under S because Bf(A)E = f(A)BE c f(A)E Let F be the closure of f(A)E Then F #- because f(A) #- O Furthermore, F #- E because g(A)f(A)E = and hence g(A)F = O Since F is obviously invariant under S, we have a contradiction Corollary 3.12 Let S be a set of operators of the Hilbert space E, leaving no closed subspace invariant except and E itself Let A be an operator such that AA * = A *A, AB = BA, and A *B = BA * for all B E S Then A = iiI for some complex number Ii Proof Write A = Al + iA2 where AI' A2 are hermitian and commute (e.g Al = (A + A*)/2) Apply the theorem to each one of Al and A2 to get the result Bi bl iography [Ab 62] [AbM 78] [Am 56] [Am 60] [Am 61] [APS 60] [ABP 75] [AH 59] [APS 75] [Be 58] [BGM 71] [BGV 92] R ABRAHAM, Lectures of Smale on Differential Topology, Columbia University, 1962 R ABRAHAM and J MARSDEN, Foundations of Mechanics, second edition, Benjamin-Cummings, 1978 W AMBROSE, Parallel translation of Riemannian curvature, Ann of Math 64 (1956) pp 337-363 W AMBROSE, The Cartan structural equations in classical Riemannian geometry, J Indian Math Soc 24 (1960) pp 23-75 W 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the geometry of paths, Quaterly J Math (Oxford) (1932) pp 33-42 Index A adjoint of d 330, 332, 333 admit partitions of unity 32 almost compact support 312 alternating 142 alternating product 126, 129 analytic 336 approximate solution 71 arc length 187 atlas 20 B Banachable base space 41 Bianchi identity 227 bilinear map associated with spray 99, 194 bilinear tensor 141, 214 bilinear tensor field 142 block 288 boundary 38, 315 bounded functional 300 bounded operator bracket of vector fields 115 bundle along the fiber 54 C canonical 1-form and 2-form canonical lifting 95 canonical spray 189 147 Cartan-Hadamard theorem 246 category Cauchy theorem 337 Cc-functional 300 change of variables formula for integration 295 change of variables formula for sprays 100 chart 21 Christoffel symbols 208 class 0' 7, 56 closed form 135, 281 closed graph theorem closed submanifold 24 cocycle condition 42 cokernel 512 commuting vector fields 120 compactly exact 322 compatible 21 complete 218 complex analytic 335 compressible 110, 180 connection 101, 102 constant curvature 238, 247 contraction 137 contraction lemma 13 convex neighborhood 214 corners 39 cotangent bundle 58, 146 covariant derivative 191, 195, 265 critical point 92, 182 curvature operator 233 curvature tensor 227 362 curve 87 cut-off function INDEX 196 functor 2, 57 functor of class CP 56 D G d* 277 Darboux theorem 150 decomposable 127, 128 degenerate block 288 density 304 dependence on parameters 70, 74 de Rham cohomology 322 derivation 115 derivative differentiable differential equations 65, 158 differential form 58, 122 differential of exponential map 240 direct product 49 direct sum 59 divergence 263 divergence theorem 329 divisor 339 domain of definition 78, 84, 90, 105 dual bundle 55, 58, 146 duality of higher degree forms 273 duality of vector fields and I-forms 141 Gauss lemma 214,241 Gauss theorem 330 g-distance 185 geodesic 95 geodesic flow 107 geodesically complete 218 global smoothness of flow 85 gradient 144 Green's formula 332 group manifold 163 E embedding 25 energy 184, 189, 251 exact form 135, 281, 321, 322 exact sequence 50, 52 exponential map 105, 109, 209, 240, 259 exterior derivative 125, 130 F factor bundle 51 fiber 41 fiber bundle 101 fiber product 29 finite type 62 first variation 253 flow 84,89 forms frame 43 Frobenius theorem 155 frontier 315 function 31 functional 36, 187, 300, 302, 345 H half plane 36 Hamiltonian 145 harmonic function 332 HB-morphism 177 hermitian operator 347 Hilbert bundle 176 Hilbert group 173, 177 Hilbert space 344 Hilbert trivialization 177 Hodge conditions 279 Hodge decomposition 279 Hodge star 273 Hodge theorem 273 homomorphism 165 Hopf-Rinow theorem 219 horizontal sub bundle 103 hyperplane 36 hypersurface 212 I immersion 24 implicit mapping theorem 18 index form 250 initial condition 65, 88 inner product 343 integrable vector bundle 154, 166 integral 11 integral curve 65 integral manifold 160 integration of density 305 integration of forms 302 interior 38 isometry 187, 208 isomorphism 83 INDEX isotopic 111 isotopy of tubular neighborhoods 111 J Jacobi differential equation 233 Jacobi field or lift 233, 269 Jacobi identity 116 Jacobian of exponential map 268 K kernel 52 kinetic energy 145, 189 Koszul's formalism 273 L Laplace operator 264, 267, 278 Laut left invariant 164 length 184,212 level hypersurface 212 Levi-Civita derivative 204 lie above 199 Lie algebra 164 Lie derivative 121, 138, 140 Lie group 163 Lie subalgebra 165 Lie subgroup 165 lifting 95, 199 linear differential equation 74 Lipschitz condition 66, 284 Lipschitz constant 66 Lis local coordinates 21 local flow 65 local isomorphism 13, 109 local projection 17 local representation 44, 65, 87, 96, 98, 122, 147, 189, 194, 202, 205 local smoothness 76, 78 locally closed 23 locally convex locally finite 31 M manifold 21 manifolds of maps 23 manifold with boundary 36, 297 manifold without boundary 37 mean valuer theorem 11 measure associated with a form 302 363 measure 285 metric 170 metric derivative 204, 208 metric isomorphism 187,208, 212 metric spray 207 minimal geodesic 217 modeled 21, 41, 142 momentum 148 morphism 2, 8, 22 Morse-Palais lemma 182 Moser's theorem 328 multilinear tensor field 59 N natural transformation negligible 315 non-degenerate 182 non-singular 141, 142, 149, 182 norm 343 norm of operator normal bundle 54, 109 normal chart 211 normal neighborhood 211 o one-parameter subgroup 168 operation on vector field 115 operations on vector bundles 56 operator 141, 174 orientation 261, 297 oriented chart 262 oriented volume 289 orthonormal frame 262, 266 P paracompact 131 parallel 200, 202, 208 parameter 70, 158 parametrized by arc length 187 partial derivative partition of unity 32 path lifting 222 perpendicular 343 Poincare lemma 135 Poisson bracket 146 polar coordinates 216, 269 positive definite 170 projection 17 proper domain of isotopy 111 pseudo Riemannian derivative 204 pseudo Riemannian manifold 171 364 pseudo Riemannian metric pull back 30 INDEX 170 R reduction to Hilbert group 177 refinement 31 regular 315 related vector fields 117 representation, local, see local representation residue theorem 340 Ricci tensor 232 Riemann tensor 227 Riemannian density 306 Riemannian manifold 171 Riemannian metric 170 Riemannian volume form 262 rule mapping 10 s Sard theorem 173, 286 scalar curvature 232 scaling 230 scalloped 34 second-order differential equation 96 second-order vector field 95 second variation 252 section sectional curvature 230 self dual 141 semi Riemannian 173 seminegative curvature 245 semipositive operator 348 shrinking lemma 13, 67 singular 199, 315 skew symmetric 173 spectral theorem 351 spectrum 352 sphere 212 split (injection) 16 split subspace split vector bundle 62 spray 97, 104, 189, 194, 208 standard 2-form 150 star operato.:- 273 step mapping 10 Stokes' theorem for rectangles 308 Stokes' theorem on a manifold 310 Stokes' theorem with singularities 316 subbundle 50, 54 submanifold 24 submersion 25 support 31,300,321 symmetric 142, 173 symmetric bilinear form on vector bundle 171 symplectic manifold 145 T tangent bundle 49 tangent curves 88 tangent space 26 tangent subbundle 153 tangent to tangent vector 26 Taylor expansions 257 Taylor formula 11 tensor bundle 59 tensor field 59 time dependent 65, 69 toplinear isomorphism topological vector space total space 41 total tubular neighborhood 108 transition map 42 transpose of dexpx 243 transversal 28 trivial vector bundle 43 trivializing covering 41 trivializable 62 tube 108 tubular map 108 tubular neighborhood 108, 178 u uniqueness theorem 68 v variation formula 252 variation of a curve 236 variation at end points 242 variation through geodesics 237 VB (vector bundle) equivalent 41 VB chart 44 VB morphism 44 vector field 87, 115 vector field along curve 201 vector subbundle 103 volume form 262, 265, 303 w wedge product 125 Whitney sum 59 ... Cataloging-in-Publkation Data Lang, Serge 192 7Differential and Riemannian manifolds I Serge Lang p cm - (G raduate texts in mathematics; 160) Includes bibliographical references (p ) and index ISBN- 13:... dimensional manifolds, modeled on a Banach space in general, a self-dual Banach space for pseudo Riemannian geometry, and a Hilbert space for Riemannian geometry In the standard pseudo Riemannian and Riemannian. .. concepts which are used in differential topology, differential geometry, and differential equations In differential topology, one studies for instance homotopy classes of maps and the possibility of