Research About The Properties Of Lightlike Curves And Canal Surfaces

The lightlike curves in Minkowski space and canal surfaces in Euclidean space are two main contents of this thesis.In Minkowski space, vectors are divided into three kinds, respectively called the spacelike, timelike and lightlike ones. This property is called the causal character of vectors. The curves in this space are classified into spacelike, timelike and lightlike ones according to the causal character of their tangent vectors, correspondingly. For the research of spacelike and timelike curves, we can refer to the methods of Euclidean space commonly. So the progress is more smoothly and the results comparatively more. However, the study for lightlike curves is relatively slow. The main reason is the scalar product of the tangent vectors vanishes everywhere. Thus, many traditional methods can not be generalized to the lightlike curves which hindered the study for such kind of curves to some extent. But the existence of lightlike curves is just one of the most important differences between the pseudo Riemannian and Riemannian geometry. So, it is of great significance to carry out the research of lightlike curves in Minkowski space.In Chapter 3 and 4, we discussed the lightlike curves in three and four dimensional Minkowski space. Based on a conformal invariant , the lightlike arc length parameter s is defined by ]= 1. Then, according to the relationship that the lightlike curve can be regarded as the integration curve of a cone curve, the basic theories are set up including the Frenet frame, Frenet formulas, structure functions and the relations between the structure functions and lightlike curvature, lightlike torsion. Using them, the lightlike helices, slant lightlike helices, lightlike Bertrand curves, lightlike normal curves, lightlike osculating curves and lightlike rectifying curves are studied. We obtained the necessary and sufficient conditions of a lightlike curve being one of the above mentioned curves and the specific expression forms are achieved. In 2012, Professor Young Ho Kim etc. defined the W-directional curve of a lightlike curve. Motivated by such an idea, we defined the W-directional associated curves of a lightlike curve. In particular, the Bertrand and Mannheim partner curves of a lightlike curve are classified completely by studying the ?-directional associated curves. At the same time, the self-associated curve of a lightlike curve is defined. We got that the lightlike curve and its self-associated curve accompany each other. In the fourth and fifth parts of Chapter 3, these curves are studied respectively.It is well-known that the differential geometry is constituted of two parts, which are the curve and surface theories. The curve theory is the simplest and most direct achievements among of them. In recent years, many curves in Euclidean space have been extended into those in Minkowski space. The lightlike curves in this thesis are just one part of them. Meanwhile, most of problems in surface theory have found their counterparts in Minkowski space. For instance, the ruled surfaces as a particular surface have been discussed widely in Minkowski space.In Euclidean space, the surfaces formed by a single parameter family of spheres are called canal surfaces which have been applied in many fields due to their intuitive geometric shapes. However, many basic geometric quantities have not been presented yet by surveying the existed conclusions. In order to improve the basic theories of canal surfaces and prepare previous works for studying canal surfaces in Minkowski space, we discussed canal surfaces in Euclidean 3-space in Chapter 5. First, the basic theories are established including three fundamental quantities, Gaussian curvature K, mean curvature H, the second Gaussian curvature K11 and two principal curvatures k1,k2. An important relation between the Gaussian curvature and mean curvature is arrived by-2H=Kr+-1. r Then, according to these theories, we discussed the Weingarten and linear Weingarten canal surfaces. In addition, in the fourth part of Chapter 5, some particular canal surfaces are discussed. The developable, minimal, K11 flat canal surfaces and those with constant Gaussian curvature, constant mean curvature and constant second Gaussian curvature are studied. Finally, the canal surfaces satisfying K11= K and K11= H are discussed, respectively.