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APPLIED
DIFFERENTIAL
GEOMETRY
A Modern Introduction

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APPLIED
DIFFERENTIAL
GEOMETRY
A Modern Introduction
Vladimir G IvancevicDefence Science and Technology Organisation, Australia
Tijana T IvancevicThe University of Adelaide, Australia
N E W J E R S E Y L O N D O N S I N G A P O R E B E I J I N G S H A N G H A I H O N G K O N G TA I P E I C H E N N A I
World Scientific

British Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library.
For photocopying of material in this volume, please pay a copying fee through the CopyrightClearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission tophotocopy is not required from the publisher.
ISBN-13 978-981-270-614-0ISBN-10 981-270-614-3
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 retrievalsystem now known or to be invented, without written permission from the Publisher.
Copyright 2007 by World Scientific Publishing Co. Pte. Ltd.
Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Printed in Singapore.
APPLIED DIFFERENTIAL GEOMETRYA Modern Introduction
Rhaimie - AppliedDifferential.pmd 6/8/2007, 2:41 PM1

April 19, 2007 16:57 WSPC/Book Trim Size for 9in x 6in ApplDifGeom
Dedicated to:
Nitya, Atma and Kali
v

Preface
Applied Differential Geometry: A Modern Introduction is a graduatelevelmonographic textbook. It is designed as a comprehensive introduction intomethods and techniques of modern differential geometry with its variousphysical and nonphysical applications. In some sense, it is a continuationof our previous book, Natural Biodynamics (World Scientific, 2006), whichcontains all the necessary background for comprehensive reading of the cur-rent book. While the previous book was focused on biodynamic applica-tions, the core applications of the new book are in the realm of modern theo-retical physics, mainly following its central line: EinsteinFeynmanWitten.Other applications include (among others): control theory, robotics, neu-rodynamics, psychodynamics and socioeconomical dynamics.
The book has six chapters. Each chapter contains both pure mathe-matics and related applications labelled by the word Application.
The first chapter provides a soft (plainEnglish) introduction into man-ifolds and related geometrical structures, for all the interested readers with-out the necessary background. As a snapshot illustration, at the end ofthe first chapter, a paradigm of generic differentialgeometric modelling isgiven, which is supposed to fit all abovementioned applications.
The second chapter gives technical preliminaries for development of themodern applied differential geometry. These preliminaries include: (i) clas-sical geometrical objects tensors, (ii) both classical and modern physicalobjects actions, and modern geometrical objects functors.
The third chapter develops modern manifold geometry, together with itsmain physical and nonphysical applications. This chapter is a neccessarybackground for comprehensive reading of the remaining chapters.
The fourth chapter develops modern bundle geometry, together with itsmain physical and nonphysical applications.
vii

viii Applied Differential Geometry: A Modern Introduction
The fifth chapter develops modern jet bundle geometry, together withits main applications in nonautonomous mechanics and field physics. Allmaterial in this chapter is based on the previous chapter.
The sixth chapter develops modern geometrical machinery of Feynmanspath integrals, together with their various physical and nonphysical appli-cations. For most of this chapter, only the third chapter is a neccessarybackground, assuming a basic understanding of quantum mechanics (asprovided in the abovementioned World Scientific book, Natural Biody-namics).
The book contains both an extensive Index (which allows easy connec-tions between related topics) and a number of cited references related tomodern applied differential geometry.
Our approach to dynamics of complex systems is somewhat similar tothe approach to mathematical physics used at the beginning of the 20thCentury by the two leading mathematicians: David Hilbert and John vonNeumann the approach of combining mathematical rigor with conceptualclarity, or geometrical intuition that underpins the rigor.
The intended audience includes (but is not restricted to) theoreti-cal and mathematical physicists; applied and pure mathematicians; con-trol, robotics and mechatronics engineers; computer and neural scientists;mathematically strong chemists, biologists, psychologists, sociologists andeconomists both in academia and industry.
Compared to all differentialgeometric books published so far, AppliedDifferential Geometry: A Modern Introduction has much wider variety ofboth physical and nonphysical applications. After comprehensive read-ing of this book, a reader should be able to both read and write journalpapers in such diverse fields as superstring & topological quantum field the-ory, nonlinear dynamics & control, robotics, biomechanics, neurodynamics,psychodynamics and socioeconomical dynamics.
V. IvancevicDefence Science & Technology Organisation, Australia
e-mail: [email protected]
T. IvancevicSchool of Mathematics, The University of Adelaide
e-mail: [email protected]
May, 2006

ix
Acknowledgments
The authors wish to thank Land Operations Division, Defence Science& Technology Organisation, Australia, for the support in developing theHuman Biodynamics Engine (HBE) and all HBErelated text in this mono-graph.
Finally, we express our gratitude to the World Scientific PublishingCompany, and especially to Ms. Zhang Ji and Mr. Rhaimie Wahap.

Glossary of Frequently Used Symbols
General
iff means if and only if; r.h.s means right hand side; l.h.s means left hand side; ODE means ordinary differential equation, while PDE means partial dif-ferential equation; Einsteins summation convention over repeated indices (not necessarilyone up and one down) is assumed in the whole text, unless explicitly statedotherwise.
Sets
N natural numbers;Z integers;R real numbers;C complex numbers;H quaternions;K number field of real numbers, complex numbers, or quaternions.
Maps
f : A B a function, (or map) between sets A Dom f and B Cod f ;
Ker f = f1(eB) a kernel of f ;Im f = f(A) an image of f ;
Coker f = Cod f/ Im f a cokernel of f ;Coim f = Dom f/Ker f a coimage of f ;
xi

xii Applied Differential Geometry: A Modern Introduction
X Y-f
h@
@@@R
Z?
g
a commutative diagram, requiring h = g f .
Derivatives
Ck(A,B) set of ktimes differentiable functions between sets A to B;C(A,B) set of smooth functions between sets A to B;C0(A,B) set of continuous functions between sets A to B;f (x) = df(x)dx derivative of f with respect to x;x total time derivative of x;t t partial time derivative;xi i xi partial coordinate derivative;f = tf + xif xi total time derivative of the scalar field f = f(t, xi);ut tu, ux xu, uxx x2u only in partial differential equations;Lxi xiL, Lxi xiL coordinate and velocity partial derivatives of theLagrangian function;d exterior derivative;dn coboundary operator;n boundary operator; = (g) affine LeviCivita connection on a smooth manifold M withRiemannian metric tensor g = gij ;ijk Christoffel symbols of the affine connection ;XT covariant derivative of the tensorfield T with respect to the vectorfield X, defined by means of ijk;T;xi T|xi covariant derivative of the tensorfield T with respect to thecoordinate basis {xi};T DTdt
Tdt absolute (intrinsic, or Bianchi) derivative of the tensor
field T upon the parameter t; e.g., acceleration vector is the absolute timederivative of the velocity vector, ai = vi Dv
i
dt ; note that in general,ai 6= vi this is crucial for proper definition of Newtonian force;LXT Lie derivative of the tensorfield T in direction of the vectorfieldX;[X,Y ] Lie bracket (commutator) of two vectorfields X and Y ;[F,G], or {F,G} Poisson bracket, or LiePoisson bracket, of two functionsF and G.

Glossary of Frequently Used Symbols xiii
Smooth Manifolds, Fibre Bundles and Jet Spaces
Unless otherwise specified, all manifolds M,N, ... are assumed Cksmooth,real, finitedimensional, Hausdorff, paracompact, connected and withoutboun

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