Introduction to Global Variational Geometry
The book is devoted to recent research in the global variational theory on smooth manifolds. Its main objective is an extension of the classical variational calculus on Euclidean spaces to (topologically nontrivial) finite-dimensional smooth manifolds; to
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Demeter Krupka
Introduction to Global Variational Geometry
Atlantis Studies in Variational Geometry Volume 1
Series editors Demeter Krupka, Masaryk University, Brno, Czech Republic Huafei Sun, Beijing Institute of Technology, Beijing, China
More information about this series at http://www.atlantis-press.com
Demeter Krupka
Introduction to Global Variational Geometry
Demeter Krupka Department of Mathematics and Lepage Research Institute University of Hradec Kralove Hradec Kralove Czech Republic and Beijing Institute of Technology Beijing China and La Trobe University Melbourne Australia
ISSN 2214-0700 ISBN 978-94-6239-072-0 DOI 10.2991/978-94-6239-073-7
ISSN 2214-0719 (electronic) ISBN 978-94-6239-073-7 (eBook)
Library of Congress Control Number: 2014945965 © Atlantis Press and the author 2015 This book, or any parts thereof, may not be reproduced for commercial purposes in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system known or to be invented, without prior permission from the Publisher. Printed on acid-free paper
Preface
The global variational geometry as introduced in this book is a branch of mathematics, devoted to extremal problems on the frontiers of differential geometry, global analysis, the calculus of variations, and mathematical physics. Its subject is, generally speaking, a geometric structure consisting of a smooth manifold endowed with a differential form. More specifically, by a variational structure, or a Lagrange structure, we mean in this book a pair ðY; qÞ, where Y is a smooth fibered manifold over an n-dimensional base manifold X and q a differential n-form, defined on the r-jet prolongation J r Y of Y. The forms q, satisfying a horizontality condition, are called the Lagrangians. The variational functional, associated with ðY; qÞ, is the realR valued function CX ðpÞ 3 c ! qX ðcÞ ¼ J r c*q 2 R, where CX ðpÞ is the set of sections of Y over a compact set X X, J r c is the r-jet prolongation of a section c, and J r c*q is an n-form on X, the pull-back of q by J r c. Over the past few decades the subject has developed to a self-contained theory of extremals of integral variational functionals for sections of fibered manifolds, invariance theory under transformations of underlying geometric structures, and differential equations related to them. The variational methods for the study of these functionals extended the corresponding notions of global analysis such as differentiation and integration theory on manifolds. Innovations appeared in the developments of topological methods needed for a deeper understanding of the global character of variational concepts such as equations for extremals and conservation laws. It has also become clear that the higher order variational functionals could hardly be studied without innovations in the multi-linear algebra, namely in the decomposition theory of tensors and differential forms by the trace operation. The resulting theory differs in many aspects from the cla
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