The Differential Geometry For One-Parameter Developable Surfaces Of Space Curve

In this thesis,we focus on the differential geometry properties of one-parameter devel-opable surfaces of curves in Euclidean space.By using the theory of Lagrangian or Legen-drian singularities,we give the classification of singularities for one-parameter developable surfaces associated with regular curves and Frenet type framed curves in Euclidean 3-space The classification and study of singularities of special curves in Euclidean space are classical issues of singularity theory.In 2016,S.Honda and M.Takahashi defined the Frenet type framed curve on the basis of framed curves.The particularity of the Frenet type framed curve is that it can have singularities and it has unit tangent vector with geometric meaning at this singularities.Inspired by S.Izumiya's method of studying rectifying developable sur-faces associated with regular curves,we define one-parameter developable surface family with space curves as the base curves in Euclidean 3-space.The normal vector of one-parameter developable surface falls in the normal plane of its base curve.One-parameter developable surface is an important submanifold in Euclidean 3-space.We study concretely the geo-metric properties of two types of one-parameter developable surface whose base curves are regular curve and Frenet type framed curve respectively,reveal the relationship between the singularities of one-parameter developable surface and the geometric invariants,classify the singularities for one-parameter developable surface by using the theory of singularitiesThere are four parts in this thesis.In Chapter 1,we introduce briefly the general situation for the historical development of the singularity theory from its beginning,present briefly the background related to this topic and research status in recent years.Moreover,we expound the research purpose,methods,content and structure of the full text.In Chapter 2,we present the basic notations and the mainly results used in the thesis In Chapter 3,we research the local differential geometry of one-parameter developable surfaces associated with regular curves in Euclidean 3-space.We classify the singularities for one-parameter developable surfaces associated with regular curves.At last,two examples are given to explain the main resultsIn Chapter 4,we investigate the local differential geometry of one-parameter developable surfaces of Frenet type framed curves in Euclidean 3-space.We classify the singularities for one-parameter developable surfaces of Frenet type framed curves.We also give an example to explain the main results.