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Flecnodal and LIE-Curves of Ruled Surfaces

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Flecnodal and LIE-Curves of Ruled Surfaces Flecnodal and LIE-curves of ruled surfaces Dissertation Zur Erlangung des akademischen Grades Doktor rerum naturalium (Dr. rer. nat.) vorgelegt der Fakultät Mathematik und Naturwissenschaften der Technischen Universität Dresden von Ashraf Khattab geboren am 12. März 1971 in Alexandria Gutachter: Prof. Dr. G. Weiss, TU Dresden Prof. Dr. G. Stamou, Aristoteles Universität Thessaloniki Prof. Dr. W. Hanna, Ain Shams Universität Kairo Eingereicht am: 13.07.2005 Tag der Disputation: 25.11.2005 Acknowledgements I wish to express my deep sense of gratitude to my supervisor Prof. Dr. G. Weiss for suggesting this problem to me and for his invaluable guidance and help during this research. I am very grateful for his strong interest and steady encouragement. I benefit a lot not only from his intuition and readiness for discussing problems, but also his way of approaching problems in a structured way had a great influence on me. I would like also to thank him as he is the one who led me to the study of line geometry, which gave me an access into a large store of unresolved problems. I am also grateful to Prof. Dr. W. Hanna, who made me from the beginning interested in geometry as a branch of mathematics and who gave me the opportunity to come to Germany to study under the supervision of Prof. Dr. Weiss in TU Dresden, where he himself before 40 years had made his Ph.D. study. My thanks also are due to some people, without whose help the research would probably not have been finished and published: Prof. Dr. W. Degen, who helped me to find a missed part of the unpublished work of Gerrit Bol [8], Frau Ing. Helga Mettke for the marvellous finishing and organization of the thesis, and Michaela Bercikova for her kind help in the translation from Czech language. I also take this opportunity to express my gratitude to DAAD for awarding me a scholarship that enabled me to study in Germany at the institute of Geometry, TU Dresden. I would like to use this occasion to congratulate all the members of the institute of Geometry, TU Dresden for the family and friendly atmosphere. I wish of course to thank my parents for having raised me in such a stable and loving environment, which has enabled me to come so far. Last but not least, I want to thank my friends, who helped me maintain my sanity during my stay in Germany and during my Ph.D. study, as I felt myself drawing in this over ambitious thesis. Preface Let us first consider a space curve c and the complex K(3) of all straight lines through c. Then we may select a (not necessarily unique) ruled surface Σ⊂K(3) by demanding that c ⊂Σ has certain projective or affine or Euclidean geometric properties on Σ. It even might happen that the demanded conditions can only be fulfilled, if c is somehow special. To give an example, we consider Σ⊂K(3) such that c is the (Euclidean) striction curve on Σ. It is well known that to any strip through c there still exists a one parameter set of ruled surfaces Σ such that c has this property. So the solution set {Σ} is too big and thus not interesting. But if we add the assumption that c should be an asymptotic curve, we will receive the one-parameter PIRONDINI set of surfaces Σ to the osculating strip of c only. To restrict the set of solution surfaces Σ, we also might start with two curves c1, c2 and (2) (2) the congruence K of lines hitting both curves ci and ask for surfaces Σ⊂K such that ci have certain properties on Σ. Demanding for example that Σ consists of common main normals of ci will lead to the well known BERTRAND curves problem and thus showing an example, where the demanded assumptions to Σ (and ci ) can not be fulfilled by a pair of arbitrary curves ci in general. As many of the classical geometric properties of a curve c lead to a quadratic functional along the generators of Σ, it is some how natural to start with a pair of curves c1 and c2 and construct Σ under certain assumptions on properties of with ci . Let ci depend on parameters uIii∈⊂\, then Σ is described by a correspondence (ut12(), u()t),tJ∈ ⊂ \, between the point sets of ci . A certain demanded property of Σ and ci ⊂Σ should allow to develop the functions uti (), at least in the vicinity of an arbitrarily chosen parameter value ui= ui,0 , as well as it should deliver TAYLOR-representations of ci , too. The topic we want to deal with belongs therefore to local differential geometry and in most cases the canonical representation of ci by curvature functions (with respect to the considered euclidean, affine or projective) geometry) will be the proper tool for such a treatment. In the following, projective differential geometric properties of ci ⊂Σ will be the main topic of consideration. So, this dissertation deals with local projective differential geometry of ruled surfaces of a real projective 3-space. The single chapters of this doctoral thesis collect and present the different treatments of ruled surfaces throughout history, as they were found in literature. In more detail the content is as follows: Chapter 1 is an introductory chapter with an explanation of the problem and giving basic definitions and presenting the aim of the dissertation. Chapter 2 provides the method of WILCZYNSKI and of his followers, who gave some examples for the effectiveness of that calculus. WILCZYNSKI’s general scheme in studying a given geometric surface is to set up a system of one or more linear homogenous differential equations such that the fundamental set of solutions of the system determine the surface uniquely up to projective transformations. The independent and the dependent variables, which appear in the differential equations and also in the parametric equations of the surface, can be subjected to certain transformations which leave the surface invariant and do not Preface iii change the form of the system of differential equations, but such transformations will in general change the coefficients of the differential equations. A function of the transformed coefficients and their derivatives and of the new dependent variables and their derivatives, which is equivalent to the same function of the original coefficients and variables, is called a covariant. A covariant which does not contain the dependent variables or their derivatives is called an invariant. WILCZYNSKI used a complete and independent system of invariants and covariants as the basis for each of his geometric studies. His general method of calculating the invariants and covariants is by means of the LIE-theory of continuos groups. Since the method is purely analytical, there is no certainty that the invariants and covariants will readily show their geometric significance. Chapter 3 presents the method of CECH and FUBINI for treating ruled surfaces from a projective geometric point of view. FUBINI’s method is based on studying projective differential geometric properties of surfaces by means of differential forms (see Appendix A). However, to obtain results with a geometrical interpretation by this method, it is necessary to pass from the differential forms to a system of differential equations of the type used by WILCZYNSKI. Thus, in CECH’s theory of ruled surfaces, the coefficients of the system of differential equations of WILCZYNSKI are decomposed into terms, from which the invariants of a ruled surface Σ become obvious. Chapter 4 treats the flecnode- and LIE-curves of ruled surfaces in the affine space. Using the affine geometric natural equations of a space curve c in the affine space, O. MAYER studied ruled surfaces Σ which have a given curve c as an asymptotic line, and he deduced the equations of its flecnodes. MAYER used the deduced formulae to study some special cases like the case of c being a TZITZEICA’s curve, the case when Σ has distinct planar flecnodes, and the case of two flecnodes lying in parallel planes. MAYER defines also a ‘harmonic transformation’ and applies it on an R-family of curves on Σ . Such an R-family (or cross ratio set) is a set of curves that intersect the generators of Σ in projectively coupled rows of points. Particularly, he called the R-family which contains the flecnode-, the LIE-curves and the so called involute curves the principal family of Σ. Chapter 5 deals with BOL’s treatment of ruled surfaces. G. BOL developed the projective differential geometry of ruled surfaces using a new point of view, which depends on HESSE’s correspondence principle. Using this principle BOL gave a geometric projective classification of ruled surfaces depending on its characteristic curve, which is an image of the flecnode curve under a so called hyperbolic mapping. BOL deduced the differential equations of a ruled surface firstly with respect to an arbitrary pair of curves q1, q2 , which is normalized in such a way that ()qq12,,qq1′′,2= const.≠ 0, and later on q1, q2 are specialised to the pair of flecnode curves. Applying what he called *-transformation, a process of differentiating and norming projective coordinates at the same time, BOL deduced a system of projective invariants, by which he explained some geometric concepts related to ruled surfaces like cross ratio sets of curves, the flecnode strip, the principal system of curves, the fundamental mapping and the sequel of flecnode tangents’ surfaces. Chapter 6 gathers J. KLAPKA’s [32] geometric constructions of the flecnode points and LIE- th points on the generators of a ruled surface Σ of 4 degree.
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