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When the Mathematical Sciences Research Institute was started in
the Fall of 1982, one of the programs was "non-linear partial
differential equations." A seminar was organized whose audience
consisted of graduate students of the University and mature
mathematicians who are not experts in the field. This volume
contains 18 of these lectures. An effort is made to have an
adequate Bibliography for further information. The Editor wishes to
take this opportunity to thank all the speakers and the authors of
the articles presented in this volume for their cooperation. S. S.
Chern, Editor Table of Contents Geometrical and Analytical
Questions Stuart S. Antman 1 in Nonlinear Elasticity An
Introduction to Euler's Equations Alexandre J. Chorin 31 for an
Incompressible Fluid Linearizing Flows and a Cohomology Phillip
Griffiths 37 Interpretation of Lax Equations The Ricci Curvature
Equation Richard Hamilton 47 A Walk Through Partial Differential
Fritz John 73 Equations Remarks on Zero Viscosity Limit for Tosio
Kato 85 Nonstationary Navier-Stokes Flows with Boundary Free
Boundary Problems in Mechanics Joseph B. Keller 99 The Method of
Partial Regularity as Robert V.
This book focuses on the elementary but essential problems in
Riemann-Finsler Geometry, which include a repertoire of rigidity
and comparison theorems, and an array of explicit examples,
illustrating many phenomena which admit only Finslerian
interpretations. "This book offers the most modern treatment of the
topic ..." EMS Newsletter.
Mathematics has a certain mystique, for it is pure and ex- act, yet
demands remarkable creativity. This reputation is reinforced by its
characteristic abstraction and its own in- dividual language, which
often disguise its origins in and connections with the physical
world. Publishing mathematics, therefore, requires special effort
and talent. Heinz G-tze, who has dedicated his life to scientific
pu- blishing, took up this challenge with his typical enthusi- asm.
This Festschrift celebrates his invaluable contribu- tions to the
mathematical community, many of whose leading members he counts
among his personal friends. The articles, written by mathematicians
from around the world and coming from diverse fields, portray the
important role of mathematics in our culture. Here, the reflections
of important mathematicians, often focused on the history of
mathematics, are collected, in recognition of Heinz G-tze's
life-longsupport of mathematics.
In this work, I have attempted to give a coherent exposition of the
theory of differential forms on a manifold and harmonic forms on a
Riemannian space. The concept of a current, a notion so general
that it includes as special cases both differential forms and
chains, is the key to understanding how the homology properties of
a manifold are immediately evident in the study of differential
forms and of chains. The notion of distribution, introduced by L.
Schwartz, motivated the precise definition adopted here. In our
terminology, distributions are currents of degree zero, and a
current can be considered as a differential form for which the
coefficients are distributions. The works of L. Schwartz, in
particular his beautiful book on the Theory of Distributions, have
been a very great asset in the elaboration of this work. The reader
however will not need to be familiar with these. Leaving aside the
applications of the theory, I have restricted myself to considering
theorems which to me seem essential and I have tried to present
simple and complete of these, accessible to each reader having a
minimum of mathematical proofs background. Outside of topics
contained in all degree programs, the knowledge of the most
elementary notions of general topology and tensor calculus and
also, for the final chapter, that of the Fredholm theorem, would in
principle be adequate.
In Riemannian geometry, measurements are made with both yardsticks and protractors. These tools are represented by a family of inner-products. In Riemann-Finsler geometry (or Finsler geometry for short), one is in principle equipped with only a family of Minkowski norms. So ardsticks are assigned but protractors are not. With such a limited tool kit, it is natural to wonder just how much geometry one can uncover and describe?It now appears that there is a reasonable answer. Finsler geometry encompasses a solid repertoire of rigidity and comparison theorems, most of them founded upon a fruitful analogue of the sectional curvature. There is also a bewildering array of explicit examples, illustrating many phenomena which admit only Finslerian interpretations. This book focuses on the elementary but essential items among these results. Much thought has gone into making the account a teachable one.
These notes consist of two parts: Selected in York 1) Geometry, New
1946, Topics University Notes Peter Lax. by Differential in the 2)
Lectures on Stanford Geometry Large, 1956, Notes J.W. University by
Gray. are here with no essential They reproduced change. Heinz was
a mathematician who mathema- Hopf recognized important tical ideas
and new mathematical cases. In the phenomena through special the
central idea the of a or difficulty problem simplest background is
becomes clear. in this fashion a crystal Doing geometry usually
lead serious allows this to to - joy. Hopf's great insight approach
for most of the in these notes have become the st- thematics,
topics I will to mention a of further try ting-points important
developments. few. It is clear from these notes that laid the on
Hopf emphasis po- differential Most of the results in smooth
differ- hedral geometry. whose is both t1al have understanding
geometry polyhedral counterparts, works I wish to mention and
recent important challenging. Among those of Robert on which is
much in the Connelly rigidity, very spirit R. and in - of these
notes (cf. Connelly, Conjectures questions open International of
Mathematicians, H- of gidity, Proceedings Congress sinki vol. 1,
407-414) 1978, .
When the Mathematical Sciences Research Institute was started in
the Fall of 1982, one of the programs was "non-linear partial
differential equations." A seminar was organized whose audience
consisted of graduate students of the University and mature
mathematicians who are not experts in the field. This volume
contains 18 of these lectures. An effort is made to have an
adequate Bibliography for further information. The Editor wishes to
take this opportunity to thank all the speakers and the authors of
the articles presented in this volume for their cooperation. S. S.
Chern, Editor Table of Contents Geometrical and Analytical
Questions Stuart S. Antman 1 in Nonlinear Elasticity An
Introduction to Euler's Equations Alexandre J. Chorin 31 for an
Incompressible Fluid Linearizing Flows and a Cohomology Phillip
Griffiths 37 Interpretation of Lax Equations The Ricci Curvature
Equation Richard Hamilton 47 A Walk Through Partial Differential
Fritz John 73 Equations Remarks on Zero Viscosity Limit for Tosio
Kato 85 Nonstationary Navier-Stokes Flows with Boundary Free
Boundary Problems in Mechanics Joseph B. Keller 99 The Method of
Partial Regularity as Robert V.
This book gives a treatment of exterior differential systems. It
will in clude both the general theory and various applications. An
exterior differential system is a system of equations on a manifold
defined by equating to zero a number of exterior differential
forms. When all the forms are linear, it is called a pfaffian
system. Our object is to study its integral manifolds, i. e.,
submanifolds satisfying all the equations of the system. A
fundamental fact is that every equation implies the one obtained by
exterior differentiation, so that the complete set of equations
associated to an exterior differential system constitutes a
differential ideal in the algebra of all smooth forms. Thus the
theory is coordinate-free and computations typically have an
algebraic character; however, even when coordinates are used in
intermediate steps, the use of exterior algebra helps to
efficiently guide the computations, and as a consequence the
treatment adapts well to geometrical and physical problems. A
system of partial differential equations, with any number of inde
pendent and dependent variables and involving partial derivatives
of any order, can be written as an exterior differential system. In
this case we are interested in integral manifolds on which certain
coordinates remain independent. The corresponding notion in
exterior differential systems is the independence condition:
certain pfaffian forms remain linearly indepen dent. Partial
differential equations and exterior differential systems with an
independence condition are essentially the same object."
Cartan geometries were the first examples of connections on a principal bundle. They seem to be almost unknown these days, in spite of the great beauty and conceptual power they confer on geometry. The aim of the present book is to fill the gap in the literature on differential geometry by the missing notion of Cartan connections. Although the author had in mind a book accessible to graduate students, potential readers would also include working differential geometers who would like to know more about what Cartan did, which was to give a notion of "espaces généralisés" (= Cartan geometries) generalizing homogeneous spaces (= Klein geometries) in the same way that Riemannian geometry generalizes Euclidean geometry. In addition, physicists will be interested to see the fully satisfying way in which their gauge theory can be truly regarded as geometry.
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