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Softcover ISBN:  9780821813560 
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Product Code:  MMONO/205.B 
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Softcover ISBN:  9780821813560 
Product Code:  MMONO/205 
List Price:  $52.00 
MAA Member Price:  $46.80 
AMS Member Price:  $41.60 
eBook ISBN:  9781470446307 
Product Code:  MMONO/205.E 
List Price:  $49.00 
MAA Member Price:  $44.10 
AMS Member Price:  $39.20 
Softcover ISBN:  9780821813560 
eBook ISBN:  9781470446307 
Product Code:  MMONO/205.B 
List Price:  $101.00 $76.50 
MAA Member Price:  $90.90 $68.85 
AMS Member Price:  $80.80 $61.20 

Book DetailsTranslations of Mathematical MonographsIwanami Series in Modern MathematicsVolume: 205; 2002; 209 ppMSC: Primary 53; 58
A minimal length curve joining two points in a surface is called a geodesic. One may trace the origin of the problem of finding geodesics back to the birth of calculus.
Many contemporary mathematical problems, as in the case of geodesics, may be formulated as variational problems in surfaces or in a more generalized form on manifolds. One may characterize geometric variational problems as a field of mathematics that studies global aspects of variational problems relevant in the geometry and topology of manifolds. For example, the problem of finding a surface of minimal area spanning a given frame of wire originally appeared as a mathematical model for soap films. It has also been actively investigated as a geometric variational problem. With recent developments in computer graphics, totally new aspects of the study on the subject have begun to emerge.
This book is intended to be an introduction to some of the fundamental questions and results in geometric variational problems, studying variational problems on the length of curves and the energy of maps.
The first two chapters treat variational problems of the length and energy of curves in Riemannian manifolds, with an indepth discussion of the existence and properties of geodesics viewed as solutions to variational problems. In addition, a special emphasis is placed on the facts that concepts of connection and covariant differentiation are naturally induced from the formula for the first variation in this problem, and that the notion of curvature is obtained from the formula for the second variation.
The last two chapters treat the variational problem on the energy of maps between two Riemannian manifolds and its solution, harmonic maps. The concept of a harmonic map includes geodesics and minimal submanifolds as examples. Its existence and properties have successfully been applied to various problems in geometry and topology. The author discusses in detail the existence theorem of EellsSampson, which is considered to be the most fundamental among existence theorems for harmonic maps. The proof uses the inverse function theorem for Banach spaces. It is presented to be as selfcontained as possible for easy reading.
Each chapter may be read independently, with minimal preparation for covariant differentiation and curvature on manifolds. The first two chapters provide readers with basic knowledge of Riemannian manifolds. Prerequisites for reading this book include elementary facts in the theory of manifolds and functional analysis, which are included in the form of appendices. Exercises are given at the end of each chapter.
This is the English translation of a book originally published in Japanese. It is an outgrowth of lectures delivered at Tohoku University and at the Summer Graduate Program held at the Institute for Mathematics and its Applications at the University of Minnesota. It would make a suitable textbook for advanced undergraduates and graduate students. This item will also be of interest to those working in analysis.
ReadershipAdvanced undergraduates, graduate students, and research mathematicians interested in differential geometry, global analysis, and analysis on manifolds.

Table of Contents

Chapters

Arclength of curves and geodesics

First and second variation formulas

Energy of maps and harmonic maps

Existence of harmonic maps

Appendix A. Fundamentals of the theory of manifolds and functional analysis

Prospects for contemporary mathematics


Reviews

A welcome contribution to and survey of some of the fundamental questions and results in geometric variational problems on the length of curves and the energy of maps.
Mathematical Reviews


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A minimal length curve joining two points in a surface is called a geodesic. One may trace the origin of the problem of finding geodesics back to the birth of calculus.
Many contemporary mathematical problems, as in the case of geodesics, may be formulated as variational problems in surfaces or in a more generalized form on manifolds. One may characterize geometric variational problems as a field of mathematics that studies global aspects of variational problems relevant in the geometry and topology of manifolds. For example, the problem of finding a surface of minimal area spanning a given frame of wire originally appeared as a mathematical model for soap films. It has also been actively investigated as a geometric variational problem. With recent developments in computer graphics, totally new aspects of the study on the subject have begun to emerge.
This book is intended to be an introduction to some of the fundamental questions and results in geometric variational problems, studying variational problems on the length of curves and the energy of maps.
The first two chapters treat variational problems of the length and energy of curves in Riemannian manifolds, with an indepth discussion of the existence and properties of geodesics viewed as solutions to variational problems. In addition, a special emphasis is placed on the facts that concepts of connection and covariant differentiation are naturally induced from the formula for the first variation in this problem, and that the notion of curvature is obtained from the formula for the second variation.
The last two chapters treat the variational problem on the energy of maps between two Riemannian manifolds and its solution, harmonic maps. The concept of a harmonic map includes geodesics and minimal submanifolds as examples. Its existence and properties have successfully been applied to various problems in geometry and topology. The author discusses in detail the existence theorem of EellsSampson, which is considered to be the most fundamental among existence theorems for harmonic maps. The proof uses the inverse function theorem for Banach spaces. It is presented to be as selfcontained as possible for easy reading.
Each chapter may be read independently, with minimal preparation for covariant differentiation and curvature on manifolds. The first two chapters provide readers with basic knowledge of Riemannian manifolds. Prerequisites for reading this book include elementary facts in the theory of manifolds and functional analysis, which are included in the form of appendices. Exercises are given at the end of each chapter.
This is the English translation of a book originally published in Japanese. It is an outgrowth of lectures delivered at Tohoku University and at the Summer Graduate Program held at the Institute for Mathematics and its Applications at the University of Minnesota. It would make a suitable textbook for advanced undergraduates and graduate students. This item will also be of interest to those working in analysis.
Advanced undergraduates, graduate students, and research mathematicians interested in differential geometry, global analysis, and analysis on manifolds.

Chapters

Arclength of curves and geodesics

First and second variation formulas

Energy of maps and harmonic maps

Existence of harmonic maps

Appendix A. Fundamentals of the theory of manifolds and functional analysis

Prospects for contemporary mathematics

A welcome contribution to and survey of some of the fundamental questions and results in geometric variational problems on the length of curves and the energy of maps.
Mathematical Reviews