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Marston Morse, Stewart Scott Cairns, "Critical Point Theory in Global Analysis and Differential Topology"

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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