le 27 AIPEI CHENNAI
TAIPEI - Series in Pure Mathematics Volume 27 Total Mean Curvature and Submanifolds of Finite Type Second Edition Bang-Yen Chen Michigan State University, USA World Scientific NEW JERSEY LONDON SINGAPORE BEIJING SHANGHAI HONG KONG CHENNAI
! 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 de Rham'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 XV
xvi Total Mean Curvature and Submanifolds of Finite Type 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 S 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
Contents xvii 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-type submanifolds 179 6.8 Submanifolds of finite type in Euclidean space 180 6.9 2-type spherical hypersurfaces 189 6.10 Spherical fc-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 Biharmonic hypersurfaces in pseudo-euclidean space... 231 7.4 Recent developments on biharmonic conjecture 237 7.5 Harmonic, biharmonic and /c-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 general ized biharmonic conjecture 254 7.9 Recent development on generalized biharmonic conjecture 257 7.10 Biminimal immersions 262 7.11 Biconservative immersions 271 7.12 Iterated Laplacian and polyharmonic submanifolds... 275 8. A-biharmonic and Null 2-type Submanifolds 277 8.1 (k, 6, A)-harmonic maps and submanifolds 277 8.2 Null 2-type hypersurfaces 281 8.3 Null 2-type submanifolds with parallel mean curvature. 8.4 Null 2-type submanifolds with constant mean curvature.. 285 290 8.5 Marginally trapped null 2-type submanifolds 293
xviii Total Mean Curvature and Submanifolds of Finite Type 8.6 A-biharmonic submanifolds of E 297 8.7 A-biharmonic submanifolds in Hm 298 8.8 A-biharmonic submanifolds in Sm and S 302 9. Applications of Finite 305 Type Theory 9.1 Total mean curvature and order of submanifolds 305 9.2 Conformal property of Aivol(M) 9.3 Total mean curvature and Ai, A2 309 310 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 9.7 Ai of minimal submanifolds of projective spaces 9.8 Further applications to spectral geometry 317 322 326... 9.9 Application to variational principle: fc-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 A(f> A4> + 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... 409.. 10.5 Finite type submanifolds in Sasakian manifolds 376 10.6 Legendre submanifolds satisfying AH^ XH^ = 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 10.11 Finite type submanifolds of real hyperbolic space 411 10.12 LT finite type hypersurfaces 413 Bibliography 421 Subject Index 451 Author Index 461