Handbook of differential geometry

Cover......Page 1
Date-line......Page 6
Preface......Page 7
Introduction......Page 9
List of Contributors......Page 11
Contents......Page 13
1. Differential geometry of webs (M.A. Akivis and V.V. Goldberg)......Page 15
0. Introduction......Page 17
1.1. Definitions and examples......Page 22
1.2. The structure equations of a web $W(n+1,n,r)$......Page 25
1.3. The structure equations of a web $W(3,2,r)$......Page 27
1.4. Webs $W(4,2,r)$......Page 28
1.5. Web projectors and Chern's connection......Page 29
1.6. Transversally geodesic and isoclinic webs $W(n+1,n,r)$......Page 30
1.7. Hexagonal webs $W(3,2,r)$ and $(2n+2)$-hedral $(n+1)$-webs $W(n+1,n,r)$, $n>2$......Page 32
2.1. $(n+1)$-webs and local differentiable $n$-quasigroups......Page 33
2.2. Structure of a web $W(n+1,n,r)$ and its coordinate $n$-quasigroups in a neighborhood of a point......Page 36
2.3. Structure of a web $W(4,2,r)$ and its coordinate quasigroups in a neighborhood of a point......Page 41
2.4. Local Akivis algebras of webs $W(3,2,r)$......Page 42
2.5. Comtrans structures of webs $W(n+1,n,r)$......Page 44
2.6. Canonical expansions of the equations of a local analytic quasigroup......Page 48
2.7. One-parameter subquasigroups and subloops of a local differentiable $n$-quasigroup......Page 52
2.8. Subwebs of webs $W(n+1,n,r)$......Page 55
3.1. Special classes of webs $W(3,2,r)$ and local differentiable binary quasigroups......Page 56
3.2. Special classes of webs $W(n+1,n,r)$ and local differentiable $n$-quasigroups......Page 61
3.3. Special classes of webs $W(4,2,r)$......Page 70
4.1. Plucker mapping and Segre cones......Page 74
4.2. Definition of an almost Grassmann structure......Page 75
4.3. Almost Grassmann structures and webs......Page 76
4.4. Almost Grassmannizable webs $W(d,n,r)$......Page 77
4.5. The Grassmannization and algebraization problems......Page 80
4.6. Linearizability of webs $W(d,2,1)$......Page 84
5.1. First-order $G$-structures......Page 85
5.2. $G$-structures of finite type......Page 87
5.3. Closed $G$-structures......Page 88
5.4. Closure of $G$-structures associated with parallelizable, group, and Moufang three-webs......Page 90
5.5. Closure of $G$-structures associated with Bol three-webs, and symmetric spaces......Page 92
5.6. Geodesic loops......Page 96
5.8. Canonical expansions for three-webs associated with closed $G$-structures......Page 98
5.9. Algebraizability of three-webs associated with closed $G$-structures......Page 101
5.10. Special classes of three-webs and quasigroups connected with closed $G$-structures......Page 102
6.1. The rank and the rank problems for webs......Page 107
6.2. Rank problems for webs $W(d,n,1)$......Page 108
6.3. Rank problems for webs $W(d,n,r)$......Page 112
6.4. Rank problems for webs $W(d,2,r)$......Page 113
6.5. $q$-rank, $q7.1. Four-dimensional webs $W(3,2,2)$......Page 117
7.3. Infinitesimal automorphisms of webs......Page 119
7.4. Some special geometric structures associated with a three-web......Page 120
7.5. Three-webs in a generalized Appell space......Page 122
7.7. Webs over algebras......Page 123
7.8. Nonholonomic webs......Page 124
7.10. Webs formed by submanifolds of different dimensions......Page 125
7.11. Curvilinear $(n+1)$-webs on an $n$-dimensional manifold......Page 127
7.12. Superwebs......Page 128
7.13. Homogeneous (left) Lie loops......Page 129
7.14. Nonassociative generalization of the theory of Lie groups and Lie algebras......Page 130
7.15. Projectiviry of homogeneous (left) Lie loops......Page 132
7.16. Global theory of webs......Page 133
8.1. Applications to mathematical physics......Page 134
8.2. Applications to physics......Page 136
8.4. Application of webs to the theory of holomorphic mappings......Page 138
8.5. Application of the theory of $(n+1)$-webs for studying point correspondences......Page 139
9. Unsolved problems......Page 140
References......Page 143
2. Spaces of metrics and curvature functionals (D.E. Blair)......Page 167
1. Riemannian metrics......Page 169
1.1. The space of Riemannian metrics......Page 170
1.2. Space of metrics for noncompact manifolds......Page 174
1.4. Extensions of the theory......Page 175
1.5. Subspaces of the space of metrics......Page 176
1.6. Spaces of associated metrics......Page 177
1.8. Metrics with conditions on the scalar curvature......Page 180
2. Curvature functionals......Page 181
2.1. The functional $A(g)$......Page 183
2.2. The functional $B(g)$......Page 184
2.3. The functional $D(g)$......Page 186
2.4. Weyl conformal curvature tensor......Page 187
2.5. Functionals on spaces of associated metrics......Page 188
2.6. Additional topics......Page 193
References......Page 195
3. Riemannian submanifolds (B.Y. Chen)......Page 201
1. Introduction......Page 205
2. Nash's embedding theorem and some related results......Page 206
2.1. Cartan-Janet's theorem......Page 207
2.2. Nash's embedding theorem......Page 208
2.4. Isometric immersions with prescribed Gaussian or Gauss-Kronecker curvature......Page 209
3.1. Fundamental equations......Page 210
3.2. Fundamental theorems......Page 212
3.3. Basic notions......Page 213
3.4. A general inequality......Page 214
3.5. Product immersions......Page 216
3.6. A relationship between $k$-Ricci tensor and shape operator......Page 217
3.7. Completeness of curvature surfaces......Page 218
4.1. Rigidity......Page 219
4.2. A reduction theorem......Page 220
5. Minimal submanifolds......Page 221
5.1. First and second variational formulas......Page 222
5.2. Jacobi operator, index, nullity and Killing nullity......Page 223
5.3. Minimal submanifolds of Euclidean space......Page 224
5.4. Minimal submanifolds of spheres......Page 234
5.5. Minimal submanifolds in hyperbolic space......Page 240
5.6. Gauss map of minimal surfaces......Page 241
5.7. Complete minimal submanifolds in Euclidean space with finite total curvature......Page 245
5.9. The geometry of Gauss image......Page 255
5.10. Stability and index of minimal submanifolds......Page 256
6. Submanifolds of finite type......Page 263
6.1. Spectral resolution......Page 264
6.2. Order and type of immersions......Page 265
6.3. Equivariant submanifolds as minimal submanifolds in their adjoint hyperquadrics......Page 266
6.4. Submanifolds of finite type......Page 267
7.1. Case: $c=\bar{c}$......Page 274
7.2. Case: $c\neq\bar{c}......Page 275
8.1. Parallel submanifolds in Euclidean space......Page 276
8.3. Parallel submanifolds in hyperbolic spaces......Page 278
8.5. Parallel submanifolds in quatemionic projective spaces......Page 279
9.1. Standard immersions......Page 280
9.3. Submanifolds with pointwise planar normal sections......Page 281
9.4. Submanifolds with geodesic normal sections and helical immersions......Page 282
9.5. Submanifolds whose geodesies are generic $W$-curves......Page 283
9.6. Symmetric spaces in Euclidean space with simple geodesies......Page 284
10. Hypersurfaces of real space forms......Page 285
10.2. Homogeneous hypersurfaces......Page 286
10.3. Isoparametric hypersurfaces......Page 287
10.4. Dupin hypersurfaces......Page 291
10.5. Hypersurfaces with constant mean curvature......Page 298
10.6. Hypersurfaces with constant higher order mean curvature......Page 302
10.7. Harmonic spaces and Lichnerowicz conjecture......Page 303
11.1. Cartan's theorem......Page 305
11.2. Totally geodesic submanifolds of symmetric spaces......Page 306
11.3. Stability of totally geodesic submanifolds......Page 310
11.4. Helgason's spheres......Page 313
11.5. Frankel's theorem......Page 314
12.1. Totally umbilical submanifolds of real space forms......Page 315
12.4. Totally umbilical submanifolds of the Cayley plane......Page 316
12.6. Totally umbilical submanifolds of locally symmetric spaces......Page 317
12.7. Extrinsic spheres in locally symmetric spaces......Page 318
12.8. Totally umbilical hypersurfaces......Page 319
13.1. Conformally flat hypersurfaces......Page 320
13.2. Conformally flat submanifolds......Page 322
14. Submanifolds with parallel mean curvature vector......Page 324
14.3. Surfaces with parallel mean curvature vector......Page 325
14.4. Surfaces with parallel normalized mean curvature vector......Page 326
14.5. Submanifolds satisfying additional conditions......Page 327
15.1. Basic properties of Kaehler submanifolds......Page 328
15.2. Complex space forms and Chern classes......Page 330
15.4. Einstein-Kaehler submanifolds and Kaehler submanifolds $\tilde{M}$ satisfying $Ric(X,Y) = \tilde{Ric}(X,Y)$......Page 331
15.5. Ogiue's conjectures and curvature pinching......Page 332
15.7. Parallel Kaehler submanifolds......Page 334
15.8. Symmetric and homogeneous Kaehler submanifolds......Page 335
16. Totally real and Lagrangian submanifolds of Kaehler manifolds......Page 336
16.1. Basic properties of Lagrangian submanifolds......Page 337
16.2. A vanishing theorem and its applications......Page 339
16.3. The Hopf lift of Lagrangian submanifolds of nonflat complex space forms......Page 340
16.4. Totally real minimal submanifolds of complex space forms......Page 341
16.5. Lagrangian real space form in complex space form......Page 342
16.6. Inequalities for Lagrangian submanifolds......Page 343
16.7. Riemannian and topological obstructions to Lagrangian immersions......Page 344
16.8. An inequality between scalar curvature and mean curvature......Page 345
16.9. Characterizations of parallel Lagrangian submanifolds......Page 346
16.10. Lagrangian $H$-umbilical submanifolds and Lagrangian catenoid......Page 347
16.11. Stability of Lagrangian submanifolds......Page 348
16.12. Lagrangian immersions and Maslov class......Page 349
17.1. Basic properties of $CR$-submanifolds of Kaehler manifolds......Page 350
17.3. Inequalities for $CR$-submanifolds......Page 352
17.4. $CR$-products......Page 353
17.5. Cyclic parallel $CR$-submanifolds......Page 354
17.6. Homogeneous and mixed foliate $CR$-submanifolds......Page 355
18. Slant submanifolds of Kaehler manifolds......Page 356
18.1. Basic properties of slant submanifolds......Page 357
18.3. Slant surfaces in complex space forms......Page 358
18.4. Slant surfaces and almost complex structures......Page 360
18.5. Slant spheres in complex projective spaces......Page 361
19.1. Almost complex curves......Page 363
19.2. Minimal surfaces of constant curvature in the nearly Kaehler 6-sphere......Page 365
19.3. Hopf hypersurfaces and almost complex curves......Page 366
19.4. Lagrangian submanifolds in nearly Kaehler 6-sphere......Page 367
20.1. Axiom of planes......Page 369
20.2. Axioms of spheres and of totally umbilical submanifolds......Page 370
20.4. Axiom of antiholomorphic $k$-planes......Page 371
20.6. Submanifolds contain many circles......Page 372
21.1. Rotation index and total curvature of a curve......Page 373
21.2. Total absolute curvature of Chern and Lashof......Page 374
21.3. Tight immersions......Page 376
21.4. Taut immersions......Page 378
22.1. Total mean curvature of surfaces in Euclidean 3-space......Page 381
22.3. Further results on total mean curvature for surfaces in Euclidean space......Page 383
22.4. Total mean curvature for arbitrary submanifolds and applications......Page 384
22.5. Some related results......Page 386
References......Page 389
4. Einstein metrics in dimension four (A. Derdzinski)......Page 433
0. Introduction......Page 435
PART I: BASICS......Page 436
1. Remarks on notation......Page 438
2. Preliminaries......Page 440
3. Some linear algebra......Page 447
4. Basic facts about curvature......Page 458
5. Einstein manifolds......Page 470
6. Special properties of dimension four......Page 476
7. Jensen's theorem......Page 490
8. How Jensen's theorem fails for indefinite metrics......Page 493
9. Kaehler manifolds......Page 496
10. The "algebraic" examples......Page 500
11. Connections and flatness......Page 507
12. Some constructions leading to flat connections......Page 510
13. Submanifolds......Page 515
14. The simplest classification theorems......Page 524
15. Einstein hypersurfaces in pseudo-Euclidean spaces......Page 531
16. Conformal changes of metrics......Page 541
17. Killing fields......Page 549
18. Extremal metrics on surfaces......Page 562
19. Other conformally-Einstein product metrics......Page 573
20. Riemannian Einstein 4-manifolds and mobility......Page 580
21. Degree of mobility: Possible values......Page 587
22. Einstein metrics conformal to Kaehler metrics......Page 594
23. Potentials for Kaehler-Einstein metrics......Page 598
PART II: SOME TOPOLOGICAL OBSTRUCTIONS......Page 606
24. The Ricci curvature and Bochner's theorems......Page 607
25. Curvature and characteristic numbers......Page 611
26. The Berger and Thorpe inequalities......Page 613
27. Degrees of mappings into hyperbolic manifolds......Page 614
28. Positive Ricci curvature and Myers's theorem......Page 618
29. $G$-structures and $G$-connections......Page 622
30. Spine-structures and spinor bundles......Page 625
31. Harmonic spinors and the Lichnerowicz theorem......Page 634
32. Non-Kaehler Hermitian Einstein metrics......Page 639
33. Hitchin's theorems on compact Einstein 4-manifolds......Page 640
34. The Seiberg-Witten equations and LeBrun's theorem......Page 643
36. Kaehler-Einstein metrics on compact complex surfaces......Page 646
37. Geometry of bivectors......Page 651
38. Weyl tensors acting on bivectors......Page 660
39. The Petrov-Segre classes of Weyl-tensor operators......Page 665
40. Classes and genera of Weyl tensors......Page 671
41. Locally symmetric pseudo-Riemannian Einstein 4-manifolds......Page 673
42. Some nondiagonalizable Weyl tensors......Page 677
43. Petrov's example......Page 681
44. Locally symmetric neutral metrics (sign pattern 1- +)......Page 687
45. Complex-analytic metrics and complexifications......Page 694
46. Pseudo-complex projective spaces......Page 700
47. More on Petrov's curvature types......Page 715
48. Lorentzian Einstein metrics in general relativity......Page 716
49. Curvature-homogeneity for neutral Einstein metrics......Page 717
References......Page 719
5. The Atiyah-Singer index theorem (P.B. Gilkey)......Page 723
1. Clifford algebras and spin structures......Page 725
2. Spectral theory......Page 732
3. The classical elliptic complexes......Page 739
4. Characteristic classes of vector bundles......Page 744
5. Characteristic classes of principal bundles......Page 751
6. The index theorem......Page 753
References......Page 759
6. Survey of isospeciral manifolds (C.S. Gordon)......Page 761
1. Spectral invariants......Page 763
1.1. The heat invariants......Page 764
1.2. The geodesic flow and the wave invariants......Page 765
2.2. Structure of isospectral sets......Page 767
3.1. Hat tori......Page 769
3.3. Euclidean domains......Page 770
4.1. Background on group representations......Page 771
4.2. Lie group quotients......Page 772
4.3. Sunada's technique......Page 774
4.5. How general is Sunada's technique?......Page 784
5. Use of Riemannian submersions......Page 785
References......Page 788
7. Submanifolds with parallel fundamental form (U. Lumiste)......Page 793
Introduction......Page 795
1. Moving frame in space form......Page 797
2. Submanifold, its second fundamental form, shape operator and curvature 2-forms......Page 799
3. Higher order fundamental forms......Page 802
4. Fundamental identities......Page 804
5. Parallelity condition and its first consequences......Page 805
6. Parallelity and Gauss map......Page 809
7. Parallelity and extrinsic local symmetry......Page 811
8. Semiparallelity condition and its particular cases......Page 814
10. Decomposition of semiparallel submanifolds......Page 818
11. Decomposition of parallel submanifolds......Page 822
12. Normally flat semiparallel submanifolds......Page 823
13. Decomposition of a normally flat parallel submanifold......Page 826
14. Decomposition of a normally flat 2-parallel submanifold......Page 827
15. Classification of semiparallel surfaces......Page 830
16. Classification of the parallel lines and surfaces......Page 832
17. On geometry of semiparallel surfaces......Page 835
18. Classification of 2-parallel surfaces......Page 837
19. Veronese submanifolds......Page 842
20. Classification of semiparallel 3-dimensional submanifolds......Page 846
21. Parallel $M^3$. Symmetric Segre orbits......Page 851
22. Three-dimensional 2-parallel submanifolds......Page 856
23. Complete parallel irreducible submanifolds as standardly imbedded symmetric $R$-spaces......Page 860
24. Some general theorems on higher order parallel submanifolds......Page 863
25. Normally flat higher order parallel submanifolds......Page 865
26. On $k$-parallel surfaces......Page 868
References......Page 871
8. Sphere theorems (K. Shiohama)......Page 879
0. Introduction......Page 881
1.1. Riemannian geometry......Page 882
1.2. Hausdorff distance and the Gromov convergence theorem......Page 885
1.3. Alexandrov spaces......Page 887
2. Uniqueness and finiteness theorems within bounded geometry......Page 891
3. The fundamental groups and Betti numbers for manifolds with lower curvature bound......Page 894
4. Sphere theorems for curvature lower bound and Alexandrov spaces......Page 897
5. The structure theorem for complete noncompact manifolds......Page 902
6. Sphere theorems and submanifold geometry......Page 908
References......Page 912
9. Affine differential geometry (U. Simon)......Page 919
Introduction......Page 921
1.1. Plane curve evolutions......Page 923
2.1. Affine connections and volume forms......Page 925
2.2. Riemannian and conformal structures......Page 926
2.4. Conjugate connections [75,76,105]......Page 927
2.5. Projective structures [75]......Page 928
2.6. Codazzi structures [88,15]......Page 929
2.7. Codazzi tensors on projectively flat manifolds......Page 930
2.8. Differential operators of Laplace type......Page 931
2.10. Weyl geometries......Page 932
2.11. Affine versions of Singer's theorem......Page 933
3.1. Affine immersions......Page 934
3.2. Hypersurfaces of real affine spaces......Page 935
4. Classical affine hypersurface theories......Page 942
4.1. Relative hypersurface theory - extension......Page 943
4.2. Centroaffine hypersurfaces......Page 948
4.3. Blaschke immersions......Page 949
4.4. The Euclidean normalization and relative geometry......Page 950
5.1. Extrinsic affine curvature and affine Gauss maps......Page 951
5.2. Intrinsic and conformal properties of $(M,h)$......Page 956
6. Degenerate hypersurfaces......Page 957
7. Global affine differential geometry......Page 958
7.2. Completeness in affine hypersurface theory......Page 959
7.3. Connections......Page 960
7.5. Codazzi tensors......Page 963
7.6. Global classifications of hypersurfaces......Page 964
7.7. Global existence and uniqueness of hypersurfaces......Page 967
7.8. Spectral geometry and applications......Page 968
7.10. Affine evolutions of Blaschke hypersurfaces......Page 969
8. Submanifolds of codimension greater than one......Page 970
References......Page 971
10. A survey on isoparametric hypcrsurfaces and their generalizations (G. Thorbergsson)......Page 977
1. Isoparametric hypersurfaces......Page 979
2. Dupin hypersurfaces......Page 990
3. Isoparametric submanifolds......Page 995
4. Equifocal and taut submanifolds......Page 1001
References......Page 1005
11. Curves (T. Willmore)......Page 1011
1. The definition of a curve......Page 1013
2. Curves in Euclidean space $E^3$ and on surfaces in $E^3$......Page 1014
3. Curves in $E^n$......Page 1016
4. Differential operators on manifolds......Page 1018
5. Connections......Page 1019
6. Curvature......Page 1020
7. Pseudo-Riemannian manifolds......Page 1022
8. The Laplacian operator......Page 1025
10. Riemannian immersions......Page 1027
11. Holonomy groups......Page 1029
12. A special problem in the calculus of variations......Page 1033
References......Page 1037
Author index......Page 1039
Subject Index......Page 1051