Submanifolds of. Total Mean Curvature and. Finite Type. Bang-Yen Chen. Series in Pure Mathematics Volume. Second Edition.

Size: px

Start display at page:

Download "Submanifolds of. Total Mean Curvature and. Finite Type. Bang-Yen Chen. Series in Pure Mathematics Volume. Second Edition."

Garry Fisher
5 years ago
Views:

1 le 27 AIPEI CHENNAI

2 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

3 ! Contents Foreword vii Preface xiii 1. Differentiable Manifolds Tensors Tensor algebra Exterior algebra Differentiable manifolds Vector fields and differential forms Sard's theorem and Morse's inequalities Lie groups and Lie algebras Fibre bundles Integration of differential forms Stokes' theorem Homology, cohomology and de Rham's theorem Frobenius' theorem Riemannian and Pseudo-Riemannian Manifolds Symmetric bilinear forms and scalar products Riemannian and pseudo-riemannian manifolds Levi-Civita connection Parallel transport Riemann curvature tensor Sectional, Ricci and scalar curvatures Indefinite real space forms Gradient, Hessian and Laplacian 47 XV

4 xvi Total Mean Curvature and Submanifolds of Finite Type 2.9 Lie derivative and Killing vector fields Weyl conformal curvature tensor Hodge Theory and Spectral Geometry * Operators d, and S Hodge-Laplace operator Elliptic differential operators Hodge-de Rham decomposition and its... applications Heat equation and its fundamental solution Spectra of some important Riemannian manifolds Spectra of flat tori Heat equation and Jacobi's elliptic functions Submanifolds Cartan-Janet's and Nash's embedding theorems Formulas of Gauss and Weingarten Shape operator of submanifolds Equations of Gauss, Codazzi and Ricci Fundamental theorems of submanifolds A universal inequality for submanifolds Reduction theorem of Erbacher-Magid Two basic formulas for submanifolds Totally geodesic submanifolds Parallel submanifolds Totally umbilical submanifolds Pseudo-umbilical submanifolds Minimal Lorentzian surfaces Cartan's structure equations Total Mean Curvature Introduction Total absolute curvature of Chern and Lashof Willmore's conjecture and Marques-Neves' theorem Total mean curvature and conformal invariants Total mean curvature for arbitrary submanifolds A variational problem on total mean curvature Surfaces in Em which are conformally equivalent to flat surfaces 140

5 Contents xvii 5.8 Total mean curvatures for surfaces in E Normal curvature and total mean curvature of surfaces Submanifolds of Finite Type Introduction Order and type of submanifolds and maps Minimal polynomial criterion A variational minimal principle Finite type immersions of homogeneous spaces Curves of finite type Classification of 1-type submanifolds Submanifolds of finite type in Euclidean space type spherical hypersurfaces Spherical fc-type hypersurfaces with k > Finite type hypersurfaces in hyperbolic space type spherical surfaces of higher codimension Biharmonic Submanifolds and Biharmonic Conjectures Necessary and sufficient conditions Biharmonic curves and surfaces in pseudo-euclidean space Biharmonic hypersurfaces in pseudo-euclidean space Recent developments on biharmonic conjecture Harmonic, biharmonic and /c-biharmonic maps Equations of biharmonic hypersurfaces Biharmonic submanifolds in sphere Biharmonic submanifolds in hyperbolic space and general ized biharmonic conjecture Recent development on generalized biharmonic conjecture Biminimal immersions Biconservative immersions Iterated Laplacian and polyharmonic submanifolds A-biharmonic and Null 2-type Submanifolds (k, 6, A)-harmonic maps and submanifolds Null 2-type hypersurfaces Null 2-type submanifolds with parallel mean curvature. 8.4 Null 2-type submanifolds with constant mean curvature Marginally trapped null 2-type submanifolds 293

6 xviii Total Mean Curvature and Submanifolds of Finite Type 8.6 A-biharmonic submanifolds of E A-biharmonic submanifolds in Hm A-biharmonic submanifolds in Sm and S Applications of Finite 305 Type Theory 9.1 Total mean curvature and order of submanifolds Conformal property of Aivol(M) 9.3 Total mean curvature and Ai, A Total mean curvature and circumscribed radii Spectra of spherical submanifolds The first standard imbedding of projective spaces 9.7 Ai of minimal submanifolds of projective spaces 9.8 Further applications to spectral geometry Application to variational principle: fc-minimality Applications to smooth maps Application to Gauss map via topology Linearly independence and orthogonal maps Adjoint hyperquadrics and orthogonal immersions = Submanifolds satisfying A(f> A4> + B Submanifolds of restricted type Additional Topics in Finite Type Theory Pointwise finite type maps Submanifolds with finite type Gauss map Submanifolds with pointwise 1-type Gauss map Submanifolds with finite type spherical Gauss map Finite type submanifolds in Sasakian manifolds Legendre submanifolds satisfying AH^ XH^ = Geometry of tensor product immersions Finite type quadric and cubic representations Finite type submanifolds of complex projective space Finite type submanifolds of complex hyperbolic space Finite type submanifolds of real hyperbolic space LT finite type hypersurfaces 413 Bibliography 421 Subject Index 451 Author Index 461

Differential Geometry of Warped Product. and Submanifolds. Bang-Yen Chen. Differential Geometry of Warped Product Manifolds. and Submanifolds.

Differential Geometry of Warped Product Manifolds and Submanifolds A warped product manifold is a Riemannian or pseudo- Riemannian manifold whose metric tensor can be decomposes into a Cartesian product