Basic definitions of Riemannian geometry. Langrangians and Euler-Lagrange equations. The length Lagrangian and its EL equation. Lecture 2 More basic definitions of Riemannian geometry. Parallel translation and the geodesic equation.
String Theory, Mirror Symmetry and the SYZ Conjecture are briefly discussed, and some results of the author on singularities of special Lagrangian fibrations of Calabi-Yau 3-folds are described. Lecture 3 Jacobi theory I. Cao, Huai-Dong & Yau, Shing-Tung (ds.) Advances in geometry and mathematical physics : lectures given at Geometry and Topology conference at.This paper gives a leisurely introduction to Calabi-Yau manifolds and special Lagrangian submanifolds from the differential geometric point of view, followed by a survey of recent results on singularities of special Lagrangian submanifolds, and their application to the SYZ Conjecture. It is aimed at graduate students in Geometry, String Theorists, and others wishing to learn the subject, and is designed to be fairly self-contained. It is based on lecture courses given at Nordfjordeid, Norway and MSRI, Berkeley in June and July 2001. We introduce Calabi-Yau m-folds via holonomy groups, Kahler geometry and the Calabi Conjecture, and special Lagrangian m-folds via calibrated geometry.
Lectures On Differential Geometry Yau Series Of Lectures
LafayetteIN 47907, USA e-mail: ttmQmath.purdue.edu Fax: 31 : Department of Mathematics, Peking University, Beijing, China : Department of Mathematics, National Taiwan University, Taiwan, China : Department of Mathematics, Osaka University, Toyonaka, Osaka 560, JapanPublished Vol. Yau Published 1994 Mathematics.SERIES ON UNIVERSITY MATHEMATICS Editors: W Y Hsiang: Department of Mathematics, University of California, Berkeley,: Department of Mathematics, Purdue University, W. In 1984, the authors gave a series of lectures on differential geometry in the Institute for Advanced. These lectures are published in this volume, which describes the major achievements. In 1984, the authors gave a series of lectures on differential geometry in the Institute for Advanced Studies in Princeton, USA. International Press, 1994 - Mathematics - 414 pages.
3: Classical Geometries W Y Hsiang Vol. 2: Lectures on Lie Group W Y Hsiang Vol. 8: Analytical Geometry I Vaisman Forthcoming Vol. 7 : Number Theory with Applications W C Winnie Li Vol. 6: A Concise Introductionto Calculus W Y Hsiang Vol.
QA641.C4913 1998 516.3'6-dc21 98-2203 I CIPBritish Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British LibraryFirst published 1999 Reprinted 2000 Copyriby World Scientific Publishing Co. ISBN 9810234945 ISBN 9810241828 (pbk) 1. P 0 Box 128, Farrer Road, Singapore 912805 LISA ofice: Suite IB, 1060 Main Street, River Edge, NJ 07661UK ofice: 57 Shelton Street, Covent Garden, London WC2H 9HELibrary of Congress Cataloging-in-Publication Data Chern, Shiing-Shen, 1911 Lectures on differential geometry / S.S. London Hong KongPublished by World Scientific Publishing Co. Lam CaliforniaState Polytechnic University, Pomona, USAWorld Scientific Singapore New Jersey. Chern University of California, USAK.
The final rendition of this English translation is carried out by the undersigned, under the guidance of S.S. The original Chinese text resulted from the efforts of several colleaguesa, and, in its final formb, was compiled by Wei-Huan Chen of Peking University. Printed in Singapore by World Scientific PrintersPreface The present book is a translation and a n expansion of an introductory text based on a lecture series delivered in Peking University (the People’s Republic of China) in 1980 by a renowned leader in differential geometry, S.S. In this case permission to photocopy is not required from the publisher.This book is printed on acid and chlorine free paper.
It was already alluded t o by Riemann in his famous Habilitation speech of 1854 and its relevance t o the calculus of variations was stressed by Hilbert in his 1900 Paris Lecture. It is Professor Chern’s opinion that the time is ripe for the subject of Finsler geometry to occupy a more prominent position within university curricula in basic differential geometry. Our hope is that the material in this text will provide a solid and comprehensive background for more advanced and specialized studies. It should be suitable as a text or a work of reference for a wide audience, including (but not limited to) advanced undergraduate and beginning graduate students in mathematics, as well as physicists interested in the diverse applications of differential geometry t o physics. This translation aims a t preserving, as far as possible, both the contents and style of Professor Chern’s lectures, the hallmarks of which are simplicity, directness, and economy of approach together with in-depth treatments of fundamental topics.
The last section in Chapter Five of the original Chinese text on completeness in Riemannian geometry has also been revised and reincorporated as section 7 of the new chapter, which treats completeness in Finsler geometry. Chern on Finsler geometry (Chapter Eight) has been added. In view of these developments, a new and rather lengthy chapter prepared by Kai S. Professor Chern himself, beginning with his early work in the 1940’s and in recent collaborations with David Bao, has initiated crucial steps and paved the way in this research. Some remarkable recent work has shown, however, that the more natural starting point of Riemannian geometry is the more general Finsler setting, and that many of the beautiful and deep results in the former have Finslerian counterparts.
Hence these tools have been developed and used extensively but at the same time, the importance of intrinsic objects with invariant properties under a change of coordinates, such as tangent vector fields, differential forms, etc., is also stressed. Even though local objects such as coordinates in a manifold are devoid of intrinsic meaning, local tools, such as Cartan’s exterior differential calculus and Ricci’s tensor analysis, are extremely useful in the study of manifolds. A central theme of the text is that gbbal and local problems of differential geometry are equally interesting and important. The present text will bear witness t o this immensely fruitful mathematical style. Professor Chern is well known for his masterful synthesis of deep geometrical insights and skillful calculations.
In addition, he took great pains to introduce me t o the beautiful and fascinating developments in Finsler geometry. Not only has he graciously put up with the plodding attempts of a novice, but has also, over the course of many months, provided me with generous support and guidance. As a physicist with relatively little formal training in mathematics, I take great pleasure in expressing my sincere gratitude t o Professor Chern.
Berwald from German t o English. Barbara Hoeling I owe a special word of gratitude for helping me translate an early paper by L. In many ways, our chapter on Finsler geometry may be viewed as providing an introduction t o the Bao-Chern-Shen treatise mentioned above, and the serious reader who wishes to explore the subject at greater depth is well-advised t o pursue that definitive work.Vii I would also like to thank my colleagues in both the Physics and Mathematics Departments of my home institution for their various kind acts of assistance, support, and encouragement, especially physicists John Fang and Soumya Chakravarti, mathematical physicist Martin Nakashima, and mathematicians Bernard Banks and Charles Amelin. A special note of heartfelt thanks is owed to David Bao, who has rendered inestimable help in the preparation of the new chapter, by carefully going over the drafts, offering freely expert advice, and generously providing much needed reference material. I am deeply grateful t o Professor David Bao of the University of Houston, t o Professor Zhongmin Shen of Indiana UniversityPurdue University at Indianapolis, and again t o Professor Chern, for allowing me to draw from these materials. The new chapter on Finsler geometry has relied heavily on the joint work of Bao and Chern , and preliminary drafts of an upcoming comprehensive treatise, “An Introduction to Riemann-Finsler Geometry,” by Bao, Chern and Shen, to be published by Springer-Verlag.
Gan and Jitan Lu of World Scientific for their marvelous expediency and professionalism in bringing this book into print. Moh for their efforts in the initial translation and Hu Sen, Chen Wei, A.N. The authors are indebted to Hung-Chieh Chang and T.T. Buratti, and our three boys, Nathan, Reuben, and Aaron, for their simply being part of a wonderful and supportive family. Last but not least, I wish t o thank my wife, Dr.
Geoff Simms and Andres Cardenas, who endured with good humor the incessant alterations in the draft, and who, with their insuperable skills in UT$, rendered the manuscript into its present form.Kai S.
0 kommentar(er)
