The Principal Normal Surfaces In Three Dimensional Space

The ruled surface is an important topic in differential geometry. In this thesis, using the classical methods of differential geometry, we study a kind of non-developable ruled surfaces in Euclidean 3-space, which are generated by the principal normal of the directrix line, called principal normal surfaces. The thesis is organized as follows.Charpter 1 briefly reviews the history of geometry.Charpter 2 introduces some basic concepts and theorem about curves and surfaces in Euclidean 3-space.In Charpter 3 we start with the standard equation of non-developable ruled surfaces then study a kind of invariants of non-developable surface, called structure functions, and give some formulas of these invariants. We also discuss the geometric properties of these invariants and give a kind of classification of the non-developable ruled surfaces in Euclidean 3-space with the theories of the invariants.In Charpter 4 we study the principal normal surfaces of common curves in Euclidean 3-space, and that of Mannheim curves and Bertrand curves in Euclidean 3-space. Then the equation of the principal normal surface of the striction line of a principal normal surface in Euclidean 3-space are given. At last we represent the curvature of the striction line with the curvature and torsion of the directrix.Charpter 5 is the prospect of the research.