Table of Contents
Foreword vii
Preface xiii
1 Differentiable Manifolds 1
1.1 Tensors 2
1.2 Tensor algebra 4
1.3 Exterior algebra 5
1.4 Differentiable manifolds 7
1.5 Vector fields and differential forms 9
1.6 Sard's theorem and Morse's inequalities 13
1.7 Lie groups and Lie algebras 15
1.8 Fibre bundles 16
1.9 Integration of differential forms 19
1.10 Stokes' theorem 23
1.11 Homology, cohomology and do Rhain's theorem 25
1.12 Frobenius' theorem 28
2 Riemannian and Pseudo-Riemannian Manifolds 31
2.1 Symmetric bilinear forms and scalar products 32
2.2 Riemannian and pseudo-Riemannian manifolds 33
2.3 Levi-Civita connection 35
2.4 Parallel transport 37
2.5 Riemann curvature tensor 41
2.6 Sectional, Ricci and scalar curvatures 43
2.7 Indefinite real space forms 46
2.8 Gradient, Hessian and Laplacian 47
2.9 Lie derivative and Killing vector fields 48
2.10 Weyl conformal curvature tensor 50
3 Hodge Theory and Spectral Geometry 51
3.1 Operators d, * and δ 52
3.2 Hodge-Laplace operator 55
3.3 Elliptic differential operators 57
3.4 Hodge-de Rham decomposition and its applications 59
3.5 Heat equation and its fundamental solution 61
3.6 Spectra of some important Riemannian manifolds 64
3.7 Spectra of flat tori 67
3.8 Heat equation and Jacobi's elliptic functions 68
4 Submanifolds 71
4.1 Cartan-Janet's and Nash's embedding theorems 72
4.2 Formulas of Gauss and Weingarten 74
4.3 Shape operator of submanifolds 78
4.4 Equations of Gauss, Codazzi and Ricci 80
4.5 Fundamental theorems of submanifolds 84
4.6 A universal inequality for submanifolds 84
4.7 Reduction theorem of Erbacher-Magid 86
4.8 Two basic formulas for submanifolds 88
4.9 Totally geodesic submanifolds 91
4.10 Parallel submanifolds 92
4.11 Totally umbilical submanifolds 94
4.12 Pseudo-umbilical submanifolds 100
4.13 Minimal Lorentzian surfaces 104
4.14 Cartan's structure equations 112
5 Total Mean Curvature 113
5.1 Introduction 113
5.2 Total absolute curvature of Chern and Lashof 114
5.3 Willmore's conjecture and Marques-Neves' theorem 119
5.4 Total mean curvature and conformal invariants 121
5.5 Total mean curvature for arbitrary submanifolds 124
5.6 A variational problem on total mean curvature 132
5.7 Surfaces in Em which are conformally equivalent to flat surfaces 140
5.8 Total mean curvatures for surfaces in E4 146
5.9 Normal curvature and total mean curvature of surfaces 153
6 Submanifolds of Finite Type 157
6.1 Introduction 157
6.2 Order and type of submanifolds and maps 158
6.3 Minimal polynomial criterion 161
6.4 A variational minimal principle 165
6.5 Finite type immersions of homogeneous spaces 168
6.6 Curves of finite type 170
6.7 Classification of 1-typo submanifolds 179
6.8 Submanifolds of finite type in Euclidean space 180
6.9 2-type spherical hypersurfaces 189
6.10 Spherical k-type hypersurfaces with k ≤ 3 200
6.11 Finite type hypersurfaces in hyperbolic space 204
6.12 2-type spherical surfaces of higher codimension 209
7 Biharmonic Submanifolds and Biharmonic Conjectures 219
7.1 Necessary and sufficient conditions 220
7.2 Biharmonic curves and surfaces in pseudo-Euclidean space 222
7.3 Bihaimonic hypersurfaces in pseudo-Euclidean space 231
7.4 Reccent developments on biharmonic conjecture 237
7.5 Harmonic, biharmonic and k-biharmonic maps 241
7.6 Equations of biharmonic hypersurfaces 245
7.7 Biharmonic submanifolds in sphere 248
7.8 Biharmonic submanifolds in hyperbolic space and generalized biharmonic conjecture 251
7.9 Recent development on generalized biharmonic conjecture 257
7.10 Biminimal immersions 202
7.11 Biconservative immersions 271
7.12 Iterated Laplacian and polyharmonic submanifolds 275
8 λ-biharmonic and Null 2-type Submanifolds 277
8.1 (k, l, λ)-harrnomc maps and submanifolds 277
8.2 Null 2-type hypersurfaces 281
8.3 Null 2-typc submanifolds with parallel mean curvature 285
8.4 Null 2-type submanifolds with constant, mean curvature 290
8.5 Marginally trapped null 2-type submanifolds 293
8.6 λ-biharmonic submanifolds of Esm 297
8.7 λ-biharmonic submanifolds in Hm 298
8.8 λ-biharmonic submanifolds in Sm and Sm1 302
9 Applications of Finite Type Theory 305
9.1 Total mean curvature and order of submanifolds 305
9.2 Conformal property of λ1, vol(M) 309
9.3 Total mean curvature and λ1, λ2 310
9.4 Total mean curvature and circumscribed radii 312
9.5 Spectra of spherical submanifolds 316
9.6 The first standard imbedding of projective spaces 317
9.7 λ1 of minimal submanifolds of projective spaces 322
9.8 Further applications to spectral geometry 326
9.9 Application to variational principle: k-minimality 328
9.10 Applications to smooth maps 334
9.11 Application to Gauss map via topology 336
9.12 Linearly independence and orthogonal maps 340
9.13 Adjoint hyperquadrics and orthogonal immersions 344
9.14 Submanifolds satisfying Δφ = Δφ + B 348
9.15 Submanifolds of restricted type 350
10 Additional Topics in Finite Type Theory 357
10.1 Pointwise finite type maps 357
10.2 Submanifolds with finite type Gauss map 359
10.3 Submanifolds with pointwise 1-type Gauss map 368
10.4 Submanifolds with finite type spherical Gauss map 374
10.5 Finite type submanifolds in Sasakian manifolds 376
10.6 Legendre submanifolds satisfying ΔHφ =λHφ 382
10.7 Geometry of tensor product immersions 387
10.8 Finite type quadric and cubic representations 394
10.9 Finite type submanifolds of complex projective space 401
10.10 Finite type submanifolds of complex hyperbolic space 409
10.11 Finite type submanifolds of real hyperbolic space 411
10.12 Lr infinite type hypersurfaces 413
Bibliography 421
Subject Index 451
Author Index 461
