Projective Differential Geometry Of Curves And Surfaces (Paperback)

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those of his own results which have been published in jour nals, and has also included occasional new results hitherto unpublished.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...