Total Mean Curvature and Submanifolds of Finite Type

Transcription

1 Total Mean Curvature and Submanifolds of Finite Type Second Edition

2 SERIES IN PURE MATHEMATICS* ISSN: Editor: C C Hsiung Associate Editors: S Kobayashi, I Satake, Y-T Siu, W-T Wu and M Yamaguti Part I. Monographs and Textbooks Volume 19: Volume 20: Volume 21: Volume 22: Volume 23: Volume 24: Volume 25: Volume 26: Volume 27: Part II. Lecture Notes Volume 11: Volume 12: Topics in Integral Geometry De-Lin Ren Almost Complex and Complex Structures C. C. Hsiung Structuralism and Structures Charles E Rickart Complex Variable Methods in Plane Elasticity Jian-Ke Lu Backgrounds of Arithmetic and Geometry An Introduction Radu Miron & Dan Brânzei Topics in Mathematical Analysis and Differential Geometry Nicolas K. Laos Introduction to the Theory of Complex Functions J.-K. Lu, S.-G. Zhang & S.-G. Liu Translation Generalized Quadrangles J. A. Thas, K. Thas & H. van Maldeghem Total Mean Curvature and Submanifolds of Finite Type, 2nd Edition Bang-Yen Chen Topics in Mathematical Analysis Th M Rassias (editor) A Concise Introduction to the Theory of Integration Daniel W Stroock Part III. Collected Works Selecta of D. C. Spencer Selected Papers of Errett Bishop Collected Papers of Marston Morse Volume 14: Selected Papers of Wilhelm P. A. Klingenberg Volume 15: Collected Papers of Y. Matsushima Volume 17: Selected Papers of J. L. Koszul Volume 18: Selected Papers of M. Toda M. Wadati (editor) *To view the complete list of the published volumes in the series, please visit:

3 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 TAIPEI CHENNAI

4 Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore USA office: 27 Warren Street, Suite , Hackensack, NJ UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Chen, Bang-yen. Total mean curvature and submanifolds of finite type / by Bang-Yen Chen (Michigan State University, USA). -- 2nd edition. pages cm. -- (Series in pure mathematics ; volume 27) Includes bibliographical references and index. ISBN (hardcover : alk. paper) -- ISBN (pbk. : alk. paper) 1. Submanifolds. 2. Curvature. I. Title. QA649.C '6--dc British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright 2015 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. Printed in Singapore

5 To my wife Pi-Mei and my children Bonny, Emery and Beatrice and their families in appreciation of their love and support

6 This page intentionally left blank

7 Foreword The present book Total Mean Curvature and Submanifolds of Finite Type, Second Edition by MSU Distinguished Professor Bang-Yen Chen is the 2014 new edition of the 1984 first edition with the same title. Here follows a quote from the Bulletin of the London Mathematical Society s review of the 1984 book, written together with Paul Verheyen: Just as his previous books ( Geometry of Submanifolds, Dekker, New York, 1973 and Geometry of Submanifolds and Its Applications, Science University of Tokyo, 1981), the author has written the present work as an invitation for further research on the topics that it deals with. The World Scientific Publ. Co. could hardly have taken a better start for its new Series in Pure Mathematics than by editing Professor Bang-Yen Chen s beautiful new book as its Volume 1. It is a real joy in the new edition to see how seriously indeed this invitation has been taken at heart by many geometers from all around the world and how the author s own thoughts on these topics have deepened and expanded in the mean time. And let me be permitted to consider the 5-minimal curve that is shown on the cover, which has become the logo of PADGE right from the moment that it was drawn by Professors Bang-Yen Chen and Franki Dillen, as a token of the close co-operation of Professor Chen with many members of the KU Leuven and former KU Brussel s Section of Geometry over a period of about forty years now and also as a tribute to the memory of Professor Dillen. The chapters 5 and 6 on Total Mean Curvature and on Submanifolds of Finite Type of the first edition now have turned into somewhat adapted new chapters 5 and 6 with the same title and now extra chapters 7, 8, 9 and 10 on Biharmonic Submanifolds and Biharmonic Conjectures, λ-biharmonic and Null 2-type Submanifolds, Applications of Finite Type Theory (a.o. including the presentation of a new variational principle of submanifolds vii

8 viii Total Mean Curvature and Submanifolds of Finite Type for which the curve on the cover arose as a very particular example) and Additional Topics in Finite Type Theory were added in the new edition. The chapters 1, 2, 3 and 4 on Differentiable Manifolds, Riemannian and Pseudo-Riemannian Manifolds, Hodge Theory and Spectral Geometry and Submanifolds of the new edition are reworked and where appropriate expanded versions of the first four chapters of the first edition. And, from beginning to end, the author made the effort to include many historical notes and observations which are very welcome for a better comprehensive readability as such and for well allowing to put the presented materials in their wider mathematical and scientifical contexts. Professor Bang-Yen Chen is one of the leading experts of the general geometry of submanifolds of our time. The new book very nicely presents the just listed really important topics in this field, whereas his previous book Pseudo-Riemannian Geometry, δ-invariants and Applications (World Scientific Publ. Co., Singapore, 2011) similarly presented equally important such topics related to the Chen curvatures. And for a survey of this field as a whole, see Professor Chen s contribution on Riemannian Submanifolds in the Handbook of Differential Geometry, Vol. 1 (Elsevier Science, Amsterdam, 2000, edited by Franki Dillen e.a.). The first edition of the present book, over the years, for many geometers has been their first real meeting with basic differential geometry and in particular with the geometry of submanifolds. For the coming generations of mathematicians and of scientists and of engineers, (for the side of applications thinking on the relevance per se of the geometry of the position vector field and of the submanifolds surface tensions a.o. and i.p. on the interest of various aspects of pseudo Riemannian geometry), I do hope that this would also be the case for the new edition. Anyway, for the professional mathematicians as well as for all other readers of this book, and luckily for them in a very small format, and, of course, only for whatever it may be worth, hereafter I would like to formulate a personal view on the geometry of submanifolds today. From Jacob Bronowski s The Origins of Knowledge and Imagination come the following quotes: The place of sight in human evolution is cardinal and The world of science is wholly dominated by the sense of sight, and in the same author s The Ascent of Man one finds a discussion of the classical theorem of Pythagoras as wonderful connection between the two. Euclid s Elements ( 300) concerned the state of geometry at that time as science of our environment as experienced by our visual and motoric senses, presented in the axiomatic-deductive trend which aims for security

9 Foreword ix of the mathematical activities, as protection against otherwise mostly intuitive and eventually too loose proceedings. Likely because it involves a happening taking place at infinity and because our kind is not able to see so far, the parallel postulate of the Elements planar Euclidean geometry inevitably, right away was pretty intriguing for many a scholar. Simon Stevin s De Thiende (1585) and De Beginselen der Weeghkonst (1586), respectively conclusively dealing with all operations involving real numbers in terms of calculations only involving natural numbers via the decimal system and introducing the rule of the parallelogram for the addition of vectors, paved the way for Descartes Géométrie (1637) of which the programme was to build up the whole 2D Euclidean geometry based only on the determination of the distances of pairs of points by means of the theorem of Pythagoras expressed in a 2D Cartesian co-ordinate system: for co-ordinate axes x and y enclosing an angle θ, the distance s between points (x, y) and (x + x, y + y) being given by s 2 = x cos θ x y + y 2 ; hereby the delicate role played by the axiomatical foundation of synthetic geometry basically was taken over by the geometrical foundation of the real number system, and moreover this programme works for all dimensions alike. Thereupon, the infinitesimal calculus could geometrically be developed and in turn this allowed Newton ( 1670) to analytically determine the curvature of the curves in a Euclidean plane E 2 at any of their points and Euler ( 1760) to determine the curvature behavior of the surfaces M 2 in a Euclidean space E 3 at any of their points in terms of the curvatures there of the normal sections in all tangent directions to M 2. In his Disquisitiones generalis circa superficies curvas (1827), Gauss carried over Descartes programme to the inner geometry of surfaces M 2 in E 3. The surfaces are described by curvilinear, say (u, v) co-ordinates and a geometrical structure is defined on these surfaces M 2 by their line element ds as expressed by the infinitesimal distance function which is naturally induced on these surfaces M 2 from the standard theorem of Pythagoras Euclidean distance function of the ambient space E 3, i.e. via a generalised theorem of Pythagoras on M 2, namely, ds 2 is given by a general homogeneous quadratic polynomial in infinitesimal changes of the curvilinear co-ordinates: ds 2 = E du 2 + 2F du dv + G dv 2. And, Gauss could prove his Krümmung K to depend only on E, F and G, i.e. his theorema egregium stating the invariance of K under all surface isometries, after having shown that K equals the product of the principal Euler curvatures, whereupon he could make the fundamental distinction between the intrinsic and the extrinsic geometries of surfaces M 2 in E 3. In this setting finally the par-

10 x Total Mean Curvature and Submanifolds of Finite Type allel postulate problem could be resolved, since, locally, the 2D hyperbolic geometry of Lobachevsky-Bolyai is realised as the intrinsic geometry on the pseudo-spheres or tractroids which are concrete surfaces with constant K < 0 in E 3. The intimate link between the validity of this non-euclidean geometry on the one hand and the validity of Euclidean geometry itself on the other brought along the profound and quite revolutionary re-evaluation of the whole field of mathematical logic on which, till at present, we can pretty well base actual confidence when doing our mathematical doings. And in Ueber die Hypothesen, and respectively Tatsachen, welche der Geometrie zu Grunde liegen (of 1854, published in 1866, and respectively 1868) Riemann, and respectively Helmholtz essentially followed the same programme to come for arbitrary dimensions n to Riemann-Finsler geometry, and respectively proper Riemannian geometry. They both started from n-fold extended Mannigfaltigkeiten M n for their basic spaces, i.e. both used systems of local co-ordinates (x 1, x 2,..., x n ), and where Riemann then by hypothesis set off with a general Riemann-Finsler metric which, to get more explicit and to go for most possible simplicity in his further exposition, he then specialised to a positive definite metric tensor g = g hk dx h dx k, Helmholtz straightforwardly came up with a squared line element ds 2 given by such a homogeneous quadratic polynomial in the infinitesimal changes of the co-ordinates, i.e. given by a generalised theorem of Pythagoras, because he found this to be the only factual possibility to allow for reasonable measurements of distances. The intrinsic geometry of surfaces M 2 in E 3 thus having been the inspiration for the wider 2D Riemannian geometry, cfr. e.g. Vincent Borrelli e.a. s tore plat en 3D, may show how human s geometrical experiences in their space actually have influenced the consideration of abstract Riemannian spaces (M n, g) of arbitrary dimensions, Riemann and Helmholtz hereto moreover having been motivated by their reflections on physics and on human vision, respectively. Next, in his Raum und Zeit (1908), Minkowski made the extension of our natural measure of distances in space (x, y, z) to a corresponding natural measure of distances in spacetime (x, y, z; t) by an indefinite revision of the classical theorem of Pythagoras, as supported by the physical Weltpostulat as well as by our psychologically different appreciations of time and of location, namely, to the indefinite 4D Minkowski metric of index 1: ds 2 = dx 2 + dy 2 + dz 2 dt 2 ; (likely hereby the three letter combination ict showed up for the first and best time, and as Minkowski put it: Man kann danach das Wesen dieses Postulates mathematisch sehr prägnant in die magische Formel kleiden: km = i sek ). And thus could begin

11 Foreword xi the development of nd pseudo or semi Riemannian geometry along the lines of proper definite Riemannian geometry. The proper Euclidean, respectively the proper Riemannian spaces, in some sense conversely, then are the pseudo Euclidean, respectively the pseudo Riemannian spaces of index 0. Finally, the isometrical embedding theorems of Nash (1956) and of Clarke and Greene (1970) state that every abstract nd pseudo Riemannian manifold can be isometrically embedded into (n + m)d pseudo Euclidean spaces with appropriately large co-dimensions m, (Nash having done the embedding of Riemannian manifolds in Euclidean spaces). Thus, pseudo Riemannian geometry essentially is equivalent with the intrinsic geometry of the pseudo Riemannian submanifolds of pseudo Euclidean spaces, and, as such, is part of the geometry of submanifolds of pseudo Euclidean spaces which itself can be seen for arbitrary dimensions and co-dimensions to correspond in our natural imagination to the abstraction of our basic static and dynamic visual sense-experiences of the real curves and the real surfaces that we do encounter in our real worlds. In conclusion, the just sketched development of geometry may show it not to be so amiss to consider the general theory of submanifolds as the geometry of the human kind, from which perspective it moreover may be well in return to look back at Bronowski s quotes and discussion that were mentioned above by way of introduction to this sketch. Leopold Verstraelen The Research Center of the Serbian Academy of Sciences and Arts and the University of Kragujevac

12 This page intentionally left blank

13 Preface It has been known for a long time that the curvature of a closed surface is related to a topological invariant, namely, the Euler characteristic, via the well-known Gauss-Bonnet theorem. Since then it was well understood that integrals of curvature invariants play very important roles in many different aspects of Riemannian geometry, such as the index theorem, heat equation, volume of tubes and geodesic balls, submanifolds theory, spectral geometry, etc. The concept of mean curvature was first introduced by Marie-Sophie Germain (cf. [Germain (1831)]). In the early 1820s, S. Germain ( ) and S. D. Poisson ( ) applied total mean curvature to describe elastic shells. Total mean curvature for surfaces in Euclidean 3-space was investigated again in the 1920s by Blaschke s school, and later by T. J. Willmore in the middle of the 1960s. In order to understand the total mean curvature for general Euclidean submanifolds, the author introduced the notions of order and type for Euclidean submanifolds in the late 1970s. By applying such notions, he introduced the notions of finite type submanifolds and finite type maps. The study of finite type submanifolds and finite type maps provides a very natural way to relate the geometry of Riemannian manifolds with the spectral behaviors of the Riemannian manifolds via their immersions. Thus one often gets useful information on eigenvalues of a Riemannian manifold which can always be isometrically imbedded in a Euclidean space according to Nash s imbedding theorem. By showing that a certain submanifold is of k-type, one can, in principle, determine k eigenvalues of the Laplacian from the roots of its minimal polynomial. A submanifold M of a (pseudo) Euclidean is called biharmonic if each (pseudo) Euclidean coordinate function of M is a biharmonic function. xiii

14 xiv Total Mean Curvature and Submanifolds of Finite Type The study of biharmonic submanifolds was initiated by the author in the middle of the 1980s in his program of understanding the theory of finite type submanifolds. Independently, biharmonic submanifolds as biharmonic maps were also investigated by G.-Y. Jiang in his study of Euler-Lagrange s equation of bienergy functional. A long standing conjecture of the author states that minimal submanifolds are the only biharmonic submanifolds in Euclidean spaces. The study of biharmonic submanifolds is nowadays a very active research subject. In particular, since 2000 biharmonic submanifolds have been receiving a growing attention and have become a popular research subject of study with many important progresses. The major results, up to the early 1980s, on total mean curvature and finite type submanifolds were collected in the first edition of author s book Total Mean Curvature and Submanifolds of Finite Type published about 30 years ago. Since then there are numerous important developments on both subjects. In this second edition, we present numerous important new results on both subjects, including recent developments on biharmonic submanifolds and biharmonic conjectures as well as the recent solution of Willmore conjecture by F. C. Marques and A. Neves, developed after the publication of the first edition. This book attempts to strike a balance between giving detailed proofs of basic results and stating many results whose proofs would take us too far afield. The author also made the effort to include many historical notes and observations for a better comprehensive readability. It is the author s hope that the readers will find this second edition a good introduction to both subjects of total mean curvature and finite type theory; providing the necessary background as well as a useful reference to recent and further research of both subjects. In concluding the preface, the author would like to thank World Scientific Publishing for the invitation to undertake this project. He also would like to express his many appreciation to Professors David E. Blair, Ivko Dimitric, Oscar J. Garay, Shun Maeta, Stefano Montaldo, Ye-Lin Ou, Mira Petrović-Torga sev, Bogdan D. Suceavă, Joeri Van der Veken and Luc Vrancken for reading parts of the manuscript and offering many valuable suggestions. In particular, the author thanks Professor Leopold Verstraelen for writing an excellent foreword for this second edition. May 31, 2014 B.-Y. Chen

15 Foreword Preface Contents 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 vii xiii xv

16 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 δ 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 E m which are conformally equivalent to flat surfaces

17 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 k-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 k-biharmonic maps Equations of biharmonic hypersurfaces Biharmonic submanifolds in sphere Biharmonic submanifolds in hyperbolic space and generalized biharmonic conjecture Recent development on generalized biharmonic conjecture Biminimal immersions Biconservative immersions Iterated Laplacian and polyharmonic submanifolds λ-biharmonic and Null 2-type Submanifolds (k, l, λ)-harmonic maps and submanifolds Null 2-type hypersurfaces Null 2-type submanifolds with parallel mean curvature Null 2-type submanifolds with constant mean curvature Marginally trapped null 2-type submanifolds

18 xviii Total Mean Curvature and Submanifolds of Finite Type 8.6 λ-biharmonic submanifolds of E m s λ-biharmonic submanifolds in H m λ-biharmonic submanifolds in S m and S m Applications of Finite Type Theory Total mean curvature and order of submanifolds Conformal property of λ 1 vol(m) Total mean curvature and λ 1, λ Total mean curvature and circumscribed radii Spectra of spherical submanifolds The first standard imbedding of projective spaces λ 1 of minimal submanifolds of projective spaces Further applications to spectral geometry Application to variational principle: k-minimality Applications to smooth maps Application to Gauss map via topology Linearly independence and orthogonal maps Adjoint hyperquadrics and orthogonal immersions Submanifolds satisfying φ = Aφ + 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 H φ = λh φ 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 L r finite type hypersurfaces Bibliography 421 Subject Index 451 Author Index 461