Total Mean Curvature and Submanifolds of Finite Type (Series in Pure Mathematics)

The purpose of this book is to introduce the reader to two interesting topics in geometry which have developed over the last fifteen years, namely, total mean curvature and submanifolds of finite type. The theory of total mean curvature is the study of the integral of the n-th power of the mean curvature of a compact n-dimensional submanifold in a Euclidean m-space and its applications to other branches of mathematics. The relation of total mean curvature to analysis, geometry and topology are discussed in detail. Motivated from these studies, the author introduces and studies submanifolds of finite type in the last chapter. Some applications of such submanifolds are also given. This book is self-contained. The author hopes that the reader will be encouraged to pursue his studies beyond the confines of the present book. Front Cover......Page 1 Title......Page 4 Copyright......Page 5 Dedication......Page 6 Preface......Page 8 CONTENTS......Page 10 1. Tensors ......Page 14 2. Tensor Algebras ......Page 18 3. Exterior Algebras ......Page 20 4. Differentiable manifolds ......Page 24 5. Vector Fields and Differential Forms ......Page 28 6. Sard's Theorem and Morse's Inequalities ......Page 33 7. Fibre Bundles ......Page 36 8. Integration of Differential Forms ......Page 41 9. Homology, Cohomology and deRham's Theorem ......Page 50 10. Frobenius' Theorem ......Page 55 1. Affine Connections ......Page 59 2. Pseudo-Riemannian Manifolds ......Page 66 3. Riemannian Manifolds ......Page 69 4. Exponential Map and Normal Coordinates ......Page 75 5. Weyl Conformal Curvature Tensor ......Page 77 6. Kaehler Manifolds and Quaternionic Kaehler Manifolds ......Page 80 7. Submersions and Projective Spaces ......Page 84 1. Operators *, \delta and \Delta ......Page 91 2. Elliptic Differential Operators ......Page 98 3. Hodge-deRham Decomposition ......Page 104 4. Heat Equation and its Fundamental Solution ......Page 108 5. Spectra of Some Important Riemannian Manifolds ......Page 113 1. Induced Connections and Second Fundamental Form ......Page 122 2. Fundamental Equations and Fundamental Theorems ......Page 129 3. Submanifoldc with Flat Normal Connection ......Page 137 4. Totally Umbilical Submanifolds ......Page 141 5. Minimal Submanifolds......Page 148 6. The First Standard Imbeddings of Projective Spaces ......Page 154 8. Riemannian Submersions ......Page 180 9. Submanifolds of Kaehler Manifolds ......Page 184 1. Some Results Concerning Surfaces in R^3 ......Page 195 2. Total Mean Curvature ......Page 200 3. Conformal Invariants ......Page 216 4. A Variational Problem Concerning Total Mean Curvature ......Page 226 5. Surfaces in Rm which are Conformally Equivalent to a Flat Surface ......Page 239 6. Surfaces in R^4 ......Page 249 7. Surfaces in Real-Space-Forms ......Page 257 1. Order of Submanifolds ......Page 262 2. Submanifolds of Finite Type ......Page 268 3. Examples of 2-type Submanifolds ......Page 273 4. Characterizations of 2-type Submanifolds ......Page 282 5. Closed Curves of Finite Type ......Page 296 6. Order and Total Mean Curvature ......Page 306 7. Some Related Inequalities ......Page 313 8. Some Applications to Spectral Geometry ......Page 316 9. Spectra of Submanifolds of Rank-one Symmetric Spaces ......Page 320 10. Mass-symmetric Submanifolds ......Page 333 Bibliography ......Page 338 Author Index ......Page 354 Subject Index ......Page 360 Bang-yen Chen. Series In Pure Mathematics, Volume 1--jacket. Includes Indexes. Bibliography: P. [324]-340.