Critical Point Theory in Global Analysis and Differential Topology
| Date: 21 April 2011, 14:04
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PREFACE
From its beginning critical point theory has been concerned with
mutual relations between topology and geometric analysis, including
differential geometry. Although it may have seemed to many to have
been directed in its initial years toward applications of topology to
analysis, one now sees that the road from topology to geometric
analysis is a two-way street. Today the methods of critical point
theory enter into the foundations of almost all studies of analysis
or geometry “in the large.”
Mathematicians are finding that the study of global analysis or
differential topology requires a knowledge not only of the separate
techniques of analysis, differential geometry, topology, and algebra,
but also a deeper understanding of how these fields can join
forces.
It is the object of this book to add to this understanding in a new
way, a way that lays rigorous and revealing foundations.
The reader may be interested in diverse problems: in the Poincart
problem when n = 3 or 4, in the existence of equilibria in conservative
fields of forces, in the existence of periodic orbits, in global aspects
of Lie theory, or even in the possibility of new approaches to homology
or homotopy theory by way of critical point theory.
In the Introduction we refer to notable recent discoveries by masters
of global analysis. This book aims to reformulate and establish some
of the first theorems underlying these advances.
Among those who appear to have a major interest in the methods
here presented are mathematical physicists. Our studies of focal
points and of equilibrium points of Newtonian potentials contribute
to geometric optics and dynamics. The possibility of new global
topological attacks on quantum mechanics has been sensed by many
who are familiar with critical point theory.
This book should be understandable to a mature first-year graduate
student who has taken introductory courses in modern algebra,
analysis, and general topology. The course in algebra should have
familiarized the student with the elements of group theory and with
fields and rings. In analysis a knowledge is needed of classical implicit
function theorems and of existence theorems for ordinary differential
equations.
In an application of the theory to critical chords of compact
differentiable manifolds, given late in the book, a knowledge of how
a short minimizing geodesic arc varies with its end points is briefly
outlined and used. There are many places in the literature where the
student can find the geometric analysis needed to clarify this use of
geodesic arcs.
The reader will find the book a source of problems and fields of
study. This is true both in analysis and topology. The student whose
preference is for analysis will find, for example, several problems
at the end of $32 on “Equilibrium Points of an Electrostatic Potential.”
One whose major interest is topology will be challenged by our
treatment of the homology of differentiable manifolds without any
use of global triangulations of the manifolds. Our treatment must be
supplemented in many ways.
This book could be used for individual study or as a basis for a
graduate course...
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