PROJECTIVE DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES By ERNEST
PRESTON LANE Professor of Mathematics in the University of Chicago
THE UNIVERSITY OF CHICAGO PRESS CHICAGO ILLINOIS PREFACE Projective
Differential Geometry is largely a product of the first three
decades of the twentieth century. The theory has been developed in
five or more different languages, by three or four well-recognized
methods, in various and sundry notations, and has been published
partly in journals not readily accessible to all. The authors aim
in writing this book was to organize an exposition of these
researches. He desired to coordinate the results achieved on both
sides of the Atlantic so as to make the work of the European
geometers more readily accessible to American students, and so as
to make better known to others the accomplishments of the American
school. The author has made use of those of his own results which
have been published in jour nals, and has also included occasional
new results hitherto unpublished. Since this book was not designed
to be an exhaustive treatise no attempt was made to include in it
all existing projective differential geometry. Cer tainly, topics
already adequately treated in other books could be some what
neglected in this one. So, for example, periodic sequences of
Laplace and the theorems of permutability receive here only passing
mention be cause these subjects are extensively discussed in books
by Tzitze ica and Eisenhart. Moreover, certain things could be
neglected because they seemed to be primarily analytical rather
than geometrical in their nature by way of illustration may be
cited the calculation of complete systems of invari ants and
covariants. As toarrangement of material, it is hoped that the
order in which topics spontaneously occurred to the author may
prove to be the natural one. There is no simpler theory to begin
with than that of curves, to which Chapter I is devoted. The theory
of ruled surfaces, which occupies Chapter II, is the next simplest.
The elements of both of these theories are prerequi site for the
study of surfaces in ordinary space, which is found in Chapter III.
The subject of conjugate nets, as developed in Chapter IV, leads
easily to transformations of surfaces in Chapter V. In Chapter VI
some parts of these projective considerations are specialized so as
to show their connec tions with metric and affine geometry. In
Chapter VII the projective theory of surfaces in hyperspace is
amplified to some extent and is generalized in order to introduce
varieties of more dimensions than two. Finally, Chapter VIII
contains a number of miscellaneous topics which it seemed unwise to
exclude altogether and which are to be regarded as more or less
supple mentary. viii PREFACE Certain mathematical attainments on
the part of the reader are prerequi site to understanding this
book. Some previous acquaintance with the fun damentals of analytic
projective geometry is highly desirable, as familiarity with
homogeneous coordinates is assumed from the outset. The reader
should be acquainted with, or have constantly at hand, such a book
as Grausteins Introduction to Higher Geometry. Moreover, the reader
is sup posed on occasion to have some knowledge of differential
equations, power series, and other portions of analysis and
algebra. There is a list of exercises at the end of each chapter.
These are designed to give the readerpractice in actually working
at problems in geometry. Many of them are also intended to point
the way to further extensions of the theory that may be found in
the literature. Some of them contain re sults not previously
published. No attempt has been made to prepare a complete
bibliography. How ever, there is a working bibliography at the end
of the text, in which the references are of two kinds. Some are to
the original memoirs. Others are to the literature thought to be
most convenient for the reader...
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