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The Theory of Finslerian Laplacians and Applications Mathematics and Its Applications The Theory of Finslerian Laplacians and Applications Mathematics and Its Applications Managing Editor: M. HAZEWINKEL Centre/or Mathematics and Computer Science, Amsterdam, The Netherlands Volume 459 The Theory of Finslerian Laplacians and Applications edited by Peter L. Antonelli Department ofMathematical Sciences, University ofAlberta, Edmonton, Alberta, Canada and Bradley C. Lackey Department of Mathematical Sciences, University ofAlberta, Edmonton, Alberta, Canada SPRINGER SCIENCE+BUSINESS MEDIA, B.V. A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-94-010-6223-7 ISBN 978-94-011-5282-2 (eBook) DOI 10.1007/978-94-011-5282-2 Printed on acid-free paper AlI Rights Reserved @1998 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1998 Softcover reprint of the hardcover 1st edition 1998 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner TABLE OF CONTENTS Prologue vii Preface xxv SECTION I. Finsler Laplacians in Application Introduction to Diffusions on Finsler Manifolds 1 P.L. Antonelli and T.l. Zastawniak Density Dependent Host/Parasite Systems of Rothschild Type and Finslerian Diffusion 13 P.L. Antonelli and T.l. Zastawniak Stochastic Finsler Geometry in the Theory of Evolution by Symbiosis 33 P.L. Antonelli and T.l. Zastawniak SECTION II. Stochastic Analysis and Brownian Motion Diffusions on Finsler Manifolds 47 P.L. Antonelli and T.l. Zastawniak Stochastic Calculus on Finsler Manifolds and an Application in Biology 63 P.L. Antonelli and T.l. Zastawniak Diffusion on the Tangent and Indicatrix Bundles of a Finsler Manifold 89 P.L. Antonelli and T.l. Zastawniak v vi Antonelli and Lackey SECTION III. Stochastic Lagrange Geometry Diffusion on the Total Space of a Vector Bundle 111 D. Hrimiuc Diffusions and Laplacians on Lagrange Manifolds 123 P.L. Antonelli and D. Hrimiuc cp-Lagrange Laplacians 133 P.L. Antonelli and D. Hrimiuc SECTION IV. Mean-Value Properties of Harmonic Functions Diffusion, Laplacian and Hodge Decomposition on Finsler Spaces 141 P.L. Antonelli and T.J. Zastawniak A Mean-Value Laplacian for Finsler Spaces 151 P. Centore SECTION V. Analytical Constructions The Non-Linear Laplacian for Finsler Manifolds 187 Z. Shen A Bochner Vanishing Theorem for Elliptic Complices 199 B. Lackey A Lichnerowicz Vanishing Theorem for Finsler Spaces 227 B. Lackey A Geometric Inequality and a WeitzenbOCk Formula for Finsler Surfaces 245 D. Bao and B. Lackey Spinors on Finsler Spaces 277 F.J. Flaherty PROLOGUE 1 Early History of the Laplacian Laplace's equation was formed within one of natural philosophy's most noble pursuits: understanding the nature of gravity. Two of the main investigations of eighteenth century science were to verify Newton's inverse square law of gravitational attraction, and with that determine the shape of the Earth. Newton, himself, claimed that the Earth was flattened about the poles; the ratio of the length of the equator to a that of a typical longitude he predicted to be ~~~. The French Academy of Sciences was eager to put this prediction to the test, and in the 1730's sent expeditions to Lapland and Peru to measure the gravitational attractions. The former expedition was led by Maupertuis and Clairaut, who confirmed Newton's predicted shape, but claimed the ratio was ~ ~ (which is far less accurate than Newton's). In 1740, Maclaurin published a proof that a fluid of constant den­ sity with constant angular velocity would have an oblate spheriod as an equilibruim state. He later gave an expression for the gravitational attrac­ tion of such a body on an external particle lying on the axis of revolution. Both these results relied on geometric methods which proved of limited use as they could only be applied to such special configurations. Similar re­ sults were obtained by Clairaut about the same time, but using analytical means. These are compiled in his famous book, Theone de Ia figure de La terre (1743). The key idea previous authors failed to apply, was the use of a potential to derive the force. The concept of using a potential function was already popular in fluid dynamics. In fact, the term ''potential function" appears in Bernoulli's text, Hydrodynamica (1738). The first explicit appearance of Laplace's equation occured in Euler's paper, "Principles of the Motion of Fluids" (1752). Euler was unable to analyze the equation, but did provide some simple polynomial solutions. vii Vlll Antonelli and Lackey Little progress was made in detennining the force of gravitational attraction until 1782, when Legendre wrote the paper, "Recherches sur l'attraction des spheroldes." Oddly enough, this paper did not appear in print until 1785, the same year of Laplace's famous paper on the subject. Legendre proved the following statement: if the attraction of a solid of rev­ olution in known along the axis of revolution, then it is known for every external point. By introducing what are now called the Legendre polyno­ mials, he could compute the radial component of force based on the given data. With this, he was able to derive the potential, and compute the remaining components of the force. Inspired by Legendre's paper, Laplace considered the general problem of gravitational attraction by a body in his paper, ''Theorie des attractions des spheroides et de la figure des planetes" (1785). Without any justifica­ tion, he begins by stating that the potential outside the body must satisfy Laplace's equation (in spherical coordinates). He then assumes the poten­ tial is of the form V = ~ Uo(0, 4» + .;.s U1 (0, 4» + . ", and in effect separates the variables. The Un he wrote in terms of the Legendre polynomials; much later, Lord Kelvin dubbed these functions the spherical harmonics. Finally, he expands the function defining the boundary of his "spheroid" in terms of these harmonics, solving for the potential, entirely. Legendre and Laplace continued their study of the Legendre polyno­ mials and the spherical harmonics through the 1780's. Laplace only made one significant error in his gravitational theory: he assumed that Laplace's equation holds for the potential on the interior of the body, as well. This was corrected by Poisson in 1813, introducing what is now called Pois­ son's equation. Yet, Poisson even criticized his own derivation as not being sufficiently rigorous. Little effort was made to study the general properties of harmonic functions until Green's privately published booklet in 1828. In this he derived what we now call Green's identity. Later, Green also provided the first existence and uniqueness proof for the Dirichlet problem. His idea was to minimize the value Riemann would later call this the Dirichlet principle, although Green was the first to introduce it. As a note, Green worked in n-dimensions, radical thinking at the time, because credit for the formation of n-dimensional geometry is given to Grassmann, with Ausdehnungslehre (1844). In 1833, one of the most significant conceptual steps in potential the­ ory was taken by Lame. He pointed out in the context of the heat equation, Prologue ix that the mathematicians of his time were only analyzing the Laplacian in rectangular coordinates. He considered this of limited value and proposed that one build a coordinate system appropriate to the problem at hand, and then express the Laplacian in that system. He constructed various types of orthogonal curvilinear coordinates in ]R3, and presented the form of the Laplacian in each. In a later work, he gave the general technique for transforming the Laplacian into any such coordinate system; this procedure is the same algorithm taught in engineering science, today. Beltrami, who was quite familiar with Riemann's differential geometry, began the study of differential invariants in the 1860's. Lame, as a side result, had shown that the two expressions and retain their form under orthogonal rotation of the coordinates. Using a method he learned from Jacobi, Beltrami found the invariant form of both these objects in the context of Riemannian manifolds. The former is just gilc/!;ifr., and the later is what is now called the Laplace-Beltrami oper­ ator where we have written g = det(gjlc}. Note that the Dirichlet principle now just reads: the functions which minimize the integral of the first invariant form are those for which the second invariant form vanish. The first invariant has a canonical general­ ization to Finsler spaces; this is the approach taken by Shen, Section IV. Yet, as this form is not quadratic, the ensuing Shen Laplacian is nonlinear. Euler's work in hydrodynamics spawned another avenue of investi­ gation. Starting with the question of when one can interchange the or­ der of integration, Cauchy developed the theory of complex integration in his seminal 1814 paper, "Memoire sur la theorie des integrales definies". His approach was to introduce the Cauchy-Riemann equations as integra­ bility criteria for finding an antiderivative of a complex valued function. He showed that functions which satisfy these equations, (Briot and Bou­ quet would later call these holomorphic), have harmonic real and imaginary parts. Riemann provided the converse relationship between harmonic and x Antonelli and Lackey holomorphic functions, showing that a harmonic function naturally induces a holomorphic function. Actually, his interest in potential theory centered around Riemann surfaces and Abelian integrals. He introduced the notion of connectivity of a Riemann surface, which Betti would later generalize and which in turn Poincare would build into homology theory. This con­ nectivity he related to the genus of the surface, and used these ideas to give the values of Abelian integrals as line8l' combinations of "periods".
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