DOCSLIB.ORG

View This Volume's Front and Back Matter

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
View This Volume's Front and Back Matter http://dx.doi.org/10.1090/surv/014 MATHEMATICAL SURVEYS AMD MONOGRAPHS NUMBER 14 GEOMETRIC ASYMPTOTICS Revised Edition VICTOR GGILLEMIN AND SHLOMO STERNBERG American Mathematical Society Providence, Rhode Island AMS (MOS) Subject Classifications (1970). Primary 20-XX, 22-XX, 35-XX, 44-XX, 49-XX, 53-XX, 58-XX, 70-XX, 78-XX, 81-XX. Library of Congress Cataloging-in-Publication Data Guillemin, V, 1937- Geometric asymptotics. (Mathematical surveys; no. 14) Includes bibliographies and index. 1. Geometry, Differential. 2. Asymptotic expansions. 3. Optics, Geometrical. I. Sternberg, Shlomo, joint author. II. Title. III. Series: American Mathematical Society. Mathematical surveys; no. 14. QA649.G86 516' .36 77-8210 ISBN 0-8218-1633-0 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publi­ cation (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Manager of Editorial Ser­ vices, American Mathematical Society, P.O. Box 6248, Providence, Rhode Island 02940-6248. The owner consents to copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law, provided that a fee of $1.00 plus $.25 per page for each copy be paid directly to the Copyright Clearance Center, Inc., 27 Congress Street, Salem, Massachusetts 01970. When paying this fee please use the code 0076-5376/91 to refer to this publication. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. Reprinted and revised (June) 1990. Copyright ©1977 by the American Mathematical Society. All rights reserved. Printed in the United States of America The American Mathematical Society retains all rights except those granted to the United States Government. The paper used in this book is acid-free and fall within the guidelines established to ensure permanence and durability. @ 109876 5 43 2 95 94 93 92 91 GEOMETRIC ASYMPTOTICS To our teacher i#> **.<&? $1 I 11: Six J I. M. GEL'FAND who has been a guide and inspiration to us we dedicate this book as an expression of our love and admiration. Preface There has been a great deal of active research recently in the fields of symplectic geometry and the theory of Fourier integral operators. In symplectic geometry there has been much progress in understanding the geometry of dynamical systems, and the process of "quantization" as applied not only to the theory of dynamical systems, but also as a key tool in the analysis of group representations. Fourier integral operators have made possible a much more systematic analysis of the singularities of solutions of linear partial differential equations than existed heretofore, together with a good deal of geometric information associated with the spectra of such operators. These two subjects are modern manifestations of themes that have occupied a central position in mathematical thought for the past three hundred years—the relations between the wave and the corpuscular theories of light. The purpose of this book is to develop these themes, and present some of the recent advances, using the language of differential geometry as a unifying influence. We cannot pretend to do justice to the history of our subject, either in the short space of this preface, or in the body of the book. Yet some very brief mention of the early history is in order. The contributions of Newton and Huygens to the theory of light are known to all students. (Huygens' "envelope construction"— wherein the superposition of singularities distributed along a family of surfaces accumulates as singularity along the envelope—makes its appearance in the modern setting as a formula for the behaviour of wave front sets under functorial operations; cf. Chapter VI, equation (3.6).) In Fresnel's prize memoir of 1818, he combined Huygens' envelope construction with Young's "principle of interfer­ ence" to explain not only the rectilinear propagation of light but also diffraction effects. Thus Fresnel applied Huygens' method to the superposition of oscilla­ tions instead of disturbances and was able to calculate the diffraction caused by straight edges, small aperatures and screens. In Chapter VI we study the geometry of the superposition of "disturbances" while in Chapter VII we study the geometry of the superposition of (high frequency) oscillations. In 1828 v VI PREFACE Hamilton published his fundamental paper on geometrical optics, introducing his "characteristics", as a key tool in the study of optical instruments. It wasn't until substantially later that Hamilton realized that his method applied equally well to the study of mechanics. Hamilton's method was developed by Jacobi and has been a cornerstone of theoretical mechanics ever since. In Chapter IV we discuss the modern version of the Hamilton-Jacobi theory—the geometry of symplectic manifolds. It is interesting to note that although Hamilton was aware of the work of Fresnel, he chose to ignore it completely in his fundamental papers on geometrical optics. Nevertheless, he made a theoretical prediction in the wave theory—the possibility of "conical refraction"—which was experimentally veri­ fied soon thereafter. Unfortunately, we shall have nothing to say in this book on the issue of conical refraction (the problem of multiple characteristics in hyperbolic differential equations) but hope that some of the methods that we develop will prove useful in this connection. In 1833, Airy published a paper dealing with the behavior of light near a caustic. (A caustic is a set where geometrical optics predicts an infinite intensity of light. An object placed at a caustic will become quite hot, and hence the name.) In this paper, he introduced the functions now known as Airy functions. In Chapter VII, we show how the theory of singularities of mappings can be applied to understand and extend Airy's results. In 1887, Kelvin introduced the method of stationary phase—a technique for the asymptotic evaluation of certain types of definite integrals—in order to explain the V shaped wake that trails behind a ship as it moves across the water. The method of stationary phase, in a form expounded by Hormander, will be our principal analytical tool. We now turn to a chapter by chapter description of the contents of the book. Chapter I. Introduction. The Method of Stationary Phase The solution of the reduced wave equation A/x 4- k2[i = 0 with initial data prescribed on a hypersurface S C R3 is given by an integral of the form Jk\x-y\ / In order to see how (*) behaves in the range of large frequencies we first consider more general integrals of the form eik*{z)a(z)dz. /• The method of stationary phase, for evaluating such integrals for k large, is discussed in detail following Hormander's idea of making use of Morse's lemma. It is then applied to (*) to explain the phenomenon of shift in phase, when light rays pass through caustics. In the Appendix to this Chapter, we present a proof PREFACE vn of Morse's lemma and its extensions to the existence of polynomial normal forms for functions satisfying the Milnor condition. The technique of proof, due to Moser and Palais, will recur several times in various contexts where normal forms of different kinds of geometric objects are required. This chapter is rather short and is intended mainly to motivate the type of asymptotic developments we consider in Chapters II and VII. Chapter II. Differential Operators and Asymptotic Solutions We discuss the Luneburg-Lax-Ludwig technique for constructing asymptotic solutions for differential equations P(x,D9r) involving a large parameter T. This technique involves solving the characteristic or eikonal equation and then inductively solving a series of transport equations. We show how these can be solved using methods of symplectic geometry and in doing so also show how the characteristic and transport equation, viewed symplecticly, make sense in the vicinity of caustics even though the asymptotic expansions blow up at caustics. This enables us, in §6, to analytically "continue" our asymptotic expansions through caustics making the appropriate phase adjustments. In §7, we apply these results to the time independent Schrodinger equation and derive the Bohr- Sommerfeld quantization conditions of Keller and Maslov. We also derive an asymptotic formula for the fundamental solution of the time dependent Schro­ dinger equation and show that it can be interpreted, as Feynmann does, as the evaluation of a Feynmann integral by means of stationary phase1. Chapter III. Geometrical Optics We indicate how the techniques of the preceding chapter can be applied to optics. The point of view we emphasize here is that geometrical optics is just the study of the bicharacteristics of Maxwell's equation. In particular we develop the elementary laws of refraction and reflection from this point of view, and discuss focusing and magnification. We spend more time than the reader may think is warranted discussing Gaussian optics. However, this section is partly intended to motivate our study of the linear symplectic group a little later on. We also want to emphasize that the abstract mathematical considerations of Chapters II and IV are quite closely connected to concrete physical applications. However the treatment is not meant to be complete. There are two excellent and highly readable texts on the subject: Born and Wolf, "Principles of Physical Optics" and Luneburg, "Mathematical Theory of Optics", and the reader is referred to them for a detailed treatment.
Load more

Recommended publications
  • Ricci Measure for Some Singular Riemannian Metrics
    Math. Ann. (2016) 365:449–471 DOI 10.1007/s00208-015-1288-7 Mathematische Annalen Ricci measure for some singular Riemannian metrics John Lott1 Received: 10 April 2015 / Revised: 14 August 2015 / Published online: 4 September 2015 © Springer-Verlag Berlin Heidelberg 2015 Abstract We define the Ricci curvature, as a measure, for certain singular torsion- free connections on the tangent bundle of a manifold. The definition uses an integral formula and vector-valued half-densities. We give relevant examples in which the Ricci measure can be computed. In the time dependent setting, we give a weak notion of a Ricci flow solution on a manifold. 1 Introduction There has been much recent work about metric measure spaces with lower Ricci bounds, particularly the Ricci limit spaces that arise as measured Gromov–Hausdorff limits of smooth manifolds with a uniform lower Ricci bound. In this paper we address the question of whether one can make sense of the Ricci curvature itself on singular spaces. From one’s intuition about a two dimensional cone with total cone angle less than 2π, the Ricci curvature should exist at best as a measure. One natural approach toward a weak notion of Ricci curvature is to use an integral formula, such as the Bochner formula. The Bochner identity says that if ω1 and ω2 are smooth compactly supported 1-forms on a smooth Riemannian manifold M then ∗ ∗ ω1, Ric(ω2) = dω1, dω2+d ω1, d ω2−∇ω1, ∇ω2 dvol . (1.1) M B John Lott [email protected] 1 Department of Mathematics, University of California - Berkeley, Berkeley, CA 94720-3840, USA 123 450 J.
    [Show full text]
  • Measures on Differentiable Stacks
    MEASURES ON DIFFERENTIABLE STACKS MARIUS CRAINIC AND JOAO˜ NUNO MESTRE Abstract. We introduce and study measures and densities (geometric mea- sures) on differentiable stacks, using a rather straightforward generalization of Haefliger’s approach to leaf spaces and to transverse measures for foliations. In general we prove Morita invariance, a Stokes formula which provides rein- terpretations in terms of (Ruelle-Sullivan type) algebroid currents, and a Van Est isomorphism. In the proper case we reduce the theory to classical (Radon) measures on the underlying space, we provide explicit (Weyl-type) formulas that shed light on Weinstein’s notion of volumes of differentiable stacks; in particular, in the symplectic case, we prove the conjecture left open in Wein- stein’s “The volume of a differentiable stack” [33]. We also revisit the notion of Haar systems (and the existence of cut-off functions). Our original moti- vation comes from the study of Poisson manifolds of compact types [9, 10, 8], which provide two important examples of such measures: the affine and the Duistermaat-Heckman measures. Mathematics Subject Classification (2010). 58H05; 22A22. Keywords: Lie groupoids, differentiable stacks, transverse measures. Contents 1. Introduction 1 2. Measures on manifolds 5 3. Transverse measures 10 4. Geometric transversemeasures(transversedensities) 18 5. Intermezzo:thecaseofproperLiegroupoids 21 6. Intermezzo:thecaseofregularLiegroupoids 26 7. Intermezzo:thecaseofsymplecticgroupoids 28 8. Stokes formula and (Ruelle-Sullivan) currents 33 9. Cohomological insight and the Van Est isomorphism 36 10. Appendix: Haarsystemsandcut-offfunctions,revisited 44 arXiv:1603.01673v2 [math.DG] 13 Apr 2020 References 51 1. Introduction In this paper we study measures on differentiable stacks or, equivalently, trans- verse measures on Lie groupoids.
    [Show full text]
  • Notes on Super Math Bernstein–Deligne–Morgan
    NOTES ON SUPER MATH MOSTLY FOLLOWING BERNSTEIN–DELIGNE–MORGAN Contents References 3 1. Linear algebra 3 1.1. Super-math as the math in the braided tensor category of super vector spaces 4 1.2. The effect of the sign rule on linear algebra over the base ring k 5 1.3. Super algebras 9 1.4. Lie algebras and their enveloping algebras 10 1.5. Linear algebra on free modules over super-algebras (inner Hom, free modules and matrices) 10 1.6. Berezinian (super determinant) of free modules and automorphisms of free modules 11 1.7. The automatic extension of algebraic concepts to the super setting (“Even rules”) 15 2. Manifolds 17 2.1. Super manifolds – definitions 17 2.2. Versions: dimensional super manifolds and super-schemes 20 ∞ 2.3. Super-manifold as a functor 20 2.4. The functor of maps between two super spaces 21 2.5. Lie groups and algebraic groups 22 2.6. Sheaves 22 3. Differential Geometry 23 3.1. Vector bundles 23 3.2. (Co)tangent bundles 24 3.3. Parity change of the tangent bundle 25 4. Integration on super affine spaces 26 4.1. Integration on affine spaces 26 Date: Long Long Time Ago in a Land Far Away . 1 2 4.2. Gaussian integrals 27 4.3. Wick’s theorem 29 5. Integration on supermanifolds 30 5.1. Integration on superdomains 30 5.2. Change of variable formula on superdomains 31 5.3. Integration of densities on super manifolds 32 5.4. Summary 33 6. Super-symmetry of integrals 34 6.1.
    [Show full text]
  • MATHEMATICAL IDEAS and NOTIONS of QUANTUM FIELD THEORY 1. Generalities on Quantum Field Theory 1.1. Classical Me
    MATHEMATICALIDEASANDNOTIONSOFQUANTUMFIELDTHEORY 1. Generalitiesonquantumfieldtheory 1.1. Classicalmechanics.In classical mechanics, we study the motion of a particle. This motion is described by a (vector) function of one variable, q = q(t), representing the position of the particle as a function of time. This function must satisfy the Newton equation of motion, q¨= U ′(q), − where U the potential energy, and the mass of the particle is 1. Another way to express this law of motion is to say that q(t) must be a solution of a certain variational problem. Namely, one introduces theLagrangian q˙2 (q)= U (q) L 2 − (the difference of kinetic and potential energy), and theactionfunctional b S(q)= (q)dt L !a (for some fixed a<b). Then the law of motion can be expressed as theleastactionprinciple: q(t) must be a critical point of S on the space of all functions with given q(a)andq (b). In other words, the Newton equation is the Euler-Lagrange equation for the solution of the variational problem defined by S. Remark1.The name “least action principle” comes from the fact that in some cases (for example when U ′′ 0) the action is not only extremized but also minimized at the solution q(t). In general, however, it≤ is not the case, and the trajectory of the particle may not be a minimum, but only a saddle point of the action. Therefore, the law of motion is better formulated as the “extremal (or stationary) action principle”; this is the way we will think of it in the future.
    [Show full text]
  • Valuations with Crofton Formula and Finsler Geometry ✩
    Advances in Mathematics 210 (2007) 733–753 www.elsevier.com/locate/aim Valuations with Crofton formula and Finsler geometry ✩ Andreas Bernig Département de Mathématiques, Chemin du Musée 23, 1700 Fribourg, Switzerland Received 30 December 2005; accepted 12 July 2006 Available online 28 August 2006 Communicated by Erwin Lutwak Abstract Valuations admitting a smooth Crofton formula are studied using Geometric Measure Theory and Ru- min’s cohomology of contact manifolds. The main technical result is a current representation of a valuation with a smooth Crofton formula. A geometric interpretation of Alesker’s product is given for such valuations. As a first application in Finsler geometry, a short proof of the theorem of Gelfand–Smirnov that Crofton den- sities are projective is derived. The Holmes–Thompson volumes in a projective Finsler space are studied. It is shown that they induce in a natural way valuations and that the Alesker product of the k-dimensional and the l-dimensional Holmes–Thompson valuation is the (k + l)-dimensional Holmes–Thompson valuation. © 2006 Elsevier Inc. All rights reserved. MSC: 53C65; 53C60; 52B45 Keywords: Valuation; Holmes–Thompson volume; Crofton formula; Projective density 0. Introduction The classical Crofton formula computes the length of a curve in the plane by averaging the number of intersection points of the curve and a straight line. Higher-dimensional generaliza- tions, where straight lines are replaced by affine planes of a fixed dimension, are known under the name Linear Kinematic Formulas. These formulas were proved by Blaschke and his school. They can be used to compute the so-called intrinsic volumes of subsets of Euclidean space.
    [Show full text]
  • Notes on Tensor Analysis
    CONTRAVARIANCE, COVARIANCE, DENSITIES, AND ALL THAT: AN INFORMAL DISCUSSION ON TENSOR CALCULUS CHRIS TIEE Abstract. This collection of notes attempts to demystify some of the vagaries of tensor analysis, to emphasize the connections between the old way of doing things and the new ways, as well as to hopefully illuminate some of the more mathematically obscure aspects which turn up all the time in physics but never get any formal treatment mathematically. In short it is a summary from what I have learned from the Quest For The Holy Grail of Understanding Tensors. Contents 0. Introduction 2 0.1. Notational Conventions 3 0.2. A Brief Aside on Useful Visualizations 6 0.3. Acknowledgements 7 1. Terminological Confusions 7 1.1. Tensors vs. Tensor Fields 7 1.2. Do Vectors Have Location? 8 1.3. Invariance Under Coordinate Changes 9 1.4. Contravariance and Covariance 13 1.5. Geometric Quantities? 14 1.6. Infinitesimals vs. Linear Functionals, Volume Forms, and Rethinking Integration 15 2. Transformation Laws and Their Relation to Fancy-Schmancy Tensors 17 2.1. Transformation Laws Again 17 2.2. Fancy-Schmancy Tensors: Densitization 19 2.3. Visualization of Forms as Tensor Densities 23 3. Pseudotensors 25 3.1. The Plan 26 3.2. Recap of Orientation on Vector Spaces 27 3.3. The Pseudoscalar Algebra 28 3.4. Pseudovectors and Their Tensor Spaces 30 3.5. The Orientation Line Bundle 33 3.6. Differential Pseudoforms and Integration 35 3.7. Oriented Manifolds: Identifying Axial and Polar Quantities 39 3.8. Honest Densities and De-densitization 41 3.9.
    [Show full text]
  • Differential Geometry Andrzej Derdzinski
    Differential Geometry Andrzej Derdzinski Author address: Dept. of Mathematics, Ohio State University, Columbus, OH 43210 E-mail address: [email protected] Contents Preface xi Chapter 1. Differentiable Manifolds 1 1. Manifolds 1 Problems 2 2. Examples of manifolds 2 Problems 4 3. Differentiable mappings 5 Problems 8 4. Lie groups 12 Problems 14 Chapter 2. Tangent Vectors 17 5. Tangent and cotangent vectors 17 Problems 21 6. Vector fields 22 Problems 24 7. Lie algebras 25 Problems 27 8. The Lie algebra of a Lie group 27 Problems 30 Chapter 3. Immersions and Embeddings 33 9. The rank theorem, immersions, submanifolds 33 Problems 36 10. More on tangent vectors 39 Problems 41 11. Lie subgroups 43 Problems 44 12. Orthogonal and unitary groups 44 Problems 49 13. Orbits of Lie-group actions 51 Problems 51 14. Whitney's embedding theorem 53 Problems 55 Chapter 4. Vector Bundles 57 15. Real and complex vector bundles 57 Problems 59 16. Vector fields on the 2-sphere 60 v vi CONTENTS Problems 65 17. Operations on bundles and vector-bundle morphisms 66 Problems 69 18. Vector bundle isomorphisms and triviality 69 Problems 70 19. Subbundles of vector bundles 71 Problems 73 Chapter 5. Connections and Curvature 75 20. The curvature tensor of a connection 75 Problems 77 21. Connections in the tangent bundle 79 Problems 80 22. Parallel transport and geodesics 81 Problems 83 23. The \comma" notation for connections 84 Problems 86 24. The Ricci-Weitzenb¨ock identity 86 Problems 89 25. Variations of curves and the meaning of flatness 90 Problems 92 26.
    [Show full text]
  • Existence of Incompressible Vortex-Class Phenomena and Variational Formulation of Raleigh–Plesset Cavitation Dynamics
    Article Existence of Incompressible Vortex-Class Phenomena and Variational Formulation of Raleigh–Plesset Cavitation Dynamics Terry Eleftherios Moschandreou 1,* and Keith Christian Afas 2 1 Department of Mathematics, University of Western Ontario, London, ON N6A 5C1, Canada 2 School of Biomedical Engineering, University of Western Ontario, London, ON N6A 5C1, Canada; [email protected] * Correspondence: [email protected] or [email protected] Abstract: The following article extends a decomposition to the Navier–Stokes Equations (NSEs) demonstrated in earlier studies by corresponding author, in order to now demonstrate the existence of a vortex elliptical set inherent to the NSEs. These vortice elliptical sets are used to comment on the existence of solutions relative to the NSEs and to identify a potential manner of investigation into the classical Millennial Problem encompassed in Fefferman’s presentation. The article also presents the utilization of a recently developed versatile variational framework by both authors in order to study a related fluid-mechanics phenomena, namely the Raleigh–Plesset equations, which are ultimately obtained from the NSEs. The article develops, for the first time, a Lagrangian density functional for a closed surface which when minimized produced the Raleigh–Plesset equations. The article then proceeds with the demonstration that the Raleigh–Plesset equations may be obtained from this energy functional and identifies the energy dissipation predicted by the proposed Lagrangian density. The importance of the novel Raleigh–Plesset functional in the greater scheme of fluid mechanics is commented upon. Citation: Moschandreou, T.E.; Afas, K.C. Existence of Incompressible Vortex-Class Keywords: variational; functional; surface; free energy; fluids; navier-stokes; millennium Phenomena & Variational Formulation of Raleigh-Plesset Cavitation Dynamics.
    [Show full text]