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OBJ (Application/Pdf) THE PROJECTIVE DIFFERENTIAL GEOMETRY OF RULED SURFACES A THESIS SUBMITTED TO THE FACULTY OF ATLANTA UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE BY WILLIAM ALBERT JONES, JR. DEPARTMENT OF MATHEMATICS ATLANTA, GEORGIA AUGUST 1949 t-lii f£l ACKNOWLEDGMENTS A special expression of appreciation is due Mr. C. B. Dansby, my advisor, whose wise tolerant counsel and broad vision have been of inestimable value in making this thesis a success. The Author ii TABLE OP CONTENTS Chapter Page I. INTRODUCTION 1 1. Historical Sketch 1 2. The General Aim of this Study. 1 3. Methods of Approach... 1 II. FUNDAMENTAL CONCEPTS PRECEDING THE STUDY OP RULED SURFACES 3 1. A Linear Space of n-dimens ions 3 2. A Ruled Surface Defined 3 3. Elements of the Theory of Analytic Surfaces 3 4. Developable Surfaces. 13 III. FOUNDATIONS FOR THE THEORY OF RULED SURFACES IN Sn# 18 1. The Parametric Vector Equation of a Ruled Surface 16 2. Osculating Linear Spaces of a Ruled Surface 22 IV. RULED SURFACES IN ORDINARY SPACE, S3 25 1. The Differential Equations of a Ruled Surface 25 2. The Transformation of the Dependent Variables. 26 3. The Transformation of the Parameter 37 V. CONCLUSIONS 47 BIBLIOGRAPHY 51 üi LIST OF FIGURES Figure Page 1. A Proper Analytic Surface 5 2. The Locus of the Tangent Lines to Any Curve at a Point Px of a Surface 10 3. The Developable Surface 14 4. The Non-developable Ruled Surface 19 5. The Transformed Ruled Surface 27a CHAPTER I INTRODUCTION 1. Historical sketch.--Projective Differential Geometry- deals with the invariant differential properties of a con¬ figuration under a group of transformations. This geometry is largely a product of the first part of the twentieth century. It has been compiled in five or more different languages. Part of this work has been published in journals not readily accessible to all readers of mathematical works. Wilczynski in 1906 published in book form his projective theory of curves and of ruled surfaces in ordinary space; five years before, he had devoted about ten memoirs to this work, published in the Transactions of the American Mathematical Society. This was a significant contribution to projective differential geometry. Wilczynski also gave to the world at this time a new method in geometry and established himself as the founder of a new school of geometers. 2. The general aim of this study.—In preparing this thesis no new theories were introduced, but through careful study the author has completed much of the computation omitted by Ernest P. Lane in his work on the projective differential geometry of ruled surfaces. 3. Methods of approach.—For the benefit of the reader chapter II has been devoted to the fundamental concepts pre¬ ceding the study of ruled surfaces. It will be understood that 1 2 the study of ruled surfaces will imply the projective differen¬ tial geometry of ruled surfaces throughout this thesis. Founda¬ tions for the theory of a ruled surface in a space of n-dimen- sions will he introduced in chapter III, and a space of n- dimensions will he defined. Wilczynski's theory of ruled surfaces is based upon a consideration of the invariants and co-variants of a system of two ordinary linear homogeneous differential equations of the second order in two dependent variables, under a suitably chosen group of transformations. This analytic basis will be used in chapter IV on ruled surfaces in ordinary space. Wilczynski's notation as is used by Lane will be used through¬ out this work and some of his most fundamental geometrical results included. In conclusion, chapter V gives briefly, a summary of all the accomplishments of this study. CHAPTER II FUNDAMENTAL CONCEPTS PRECEDING THE STUDY OF RULED SURFACES 1. A linear space of n-dimensions.—A space which has the property that a straight line joining two distinct points in it lies entirely in the space is called a linear space, and is denoted by Sn. The coordinates of any point of a linear space of n-dimensions are represented by n + 1 homogeneous coordinates. A point x is represented in a space of n-dimen- sions by Px. The coordinates of the point x are (x^»x2,.. .xn<(>^) x Sometimes the set (x^,Xg,...* nV^) is thought of as a vector x with n ♦ 1 components. A scalar may be thought of as a vector with only one component. 2. A ruled surface.—Just as a curve can be thought of as the path of a moving point, a ruled surface can be defined as the locus of a moving straight line, and may be described as a single infinity of straight lines which are generators of the surface. The essential characteristic of a ruled surface is that through each point of the surface there passes one straight line that lies entirely on the surface. A quadric is an example of a ruled surface. A non-singular quadric surface in ordinary space S3 is the only example of a ruled surface having precisely two generators through every point of the surface. 3. Elements of the theory of analytic surfaces.--An 3 4 analytic surface may be defined as follows: If the n * 1 homogeneous coordinates x of a point Px in a linear space of n-dimensions are given as single-valued analytic functions of two independent variables u,v by equations of the form (1) Z x1(u, v) (i Z 1,2,3, ,n+l), then the locus of Px as u,v vary is an analytic surface S. Equations (1) are spoken of as parametric equations of the surface S. If we make x a general one of the x^ and drop the subscripts, we can replace (1) by (la) x Z x(u,v) which is called the parametric vector equation of the surface S. In order for a surface to be a proper analytic surface, the coordinates of x must satisfy no linear homogeneous partial differential equation of the first order of the form (2) ax^j +■ bxv + cx z 0, in which the subscripts indicate partial differentiation with respect to u and v, and a,b, and c are scalar functions of u, v, not all zero. If the coordinates x of a point Px are given as analytic functions of u,v and are suppose to satisfy an equation of the form of equation (2), to integrate this equa¬ tion would show that the coordinates could be expressed in the form x± : p f^t) (i r 1,2, n+1), wherein ^ is a function of u,v which is a particular solution of the equation 5 Pig. 1. A Proper Analytic Surface 6 atu + btv = °» while the f^ are functions of t. The locus of Px might de¬ generate into a fixed point or a curve. We shall confine our discussion to a proper analytic surface, and avoid cases where equation (2) may happen to be satisfied conditionally. When the parameter u in the equation x I x(u,v) remains fixed and v varies, the point Px describes a curve on a surface S for which v is the parameter called the v-curve upon which u is a constant, and hence du Z 0. This curve is denoted as Cv. Similarly when the parameter v remains fixed and u varies the locus of the point Px describes the u-curve on the surface, and is denoted as Cu. On the u-curve, v is a constant, and hence dv a 0. These curves are called the para¬ metric curves on the surface S, and the tangents to the u- curves and v-curves are called u-tangents and v-tangents re¬ spectively. All of these tangents are spoken of as parametric tangents, (See Pig. l). Two one-parameter families of curves on a surface S are said to form a net if through each point of S there passes just one curve of each family, the two tangents of the curves at the point being distinct. We shall suppose in our discussion that the parametric curves form a net, namely the parametric net, on the portion of the surface under consideration. Each pair of values of the parameters u,v locates a 7 point Px on a surface S. In a region on S where points Px and pairs of values of u,v are in continuous one-to-one corre¬ spondence, u,v are correctly called coordinates of Px. We refer to these as curvilinear coordinates to distinguished them from the projective homogeneous coordinates x. Any curve on a surface can be represented by a curvi¬ linear equation in curvilinear coordinates of the form, (3) C: <f{ u,v) = 0, and can be written parametrically by placing u z u( t), v r v( t). The slope of the tangent of curve C at the point Px is dv< du* To write the differential equation of a single infinity of curves on S through Px, set ~ = À (u,v) ( À s a scalar function of t) du we have dv I A(u,v) du, then (4) dv - A (u, v) du z 0 which is the desired equation. The equation of any net of curves on a surface S may be found as follows: Prom metric differential geometry we know that the differential of an arc of a curve may be expressed in the form of non-homogeneous coordinates as ds^ - dx^ + dy^ + dz^, 8 where x,y and z are functions of s.
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