Extrinsic Flat Surfaces Along a Curve on a Surface in the Unit Three-Sphere
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Extrinsic Flat Surfaces Along a Curve on a Surface in the Unit Three-Sphere Yang Jiang and Shyuichi Izumiya Abstract. In this paper, we consider the curves on the surface in the unit 3-sphere. For a regular curve on a surface in the unit 3-sphere, we have a moving frame along the curve which is called a spherical Darboux frame. We induce two special vector fields along the curve with respect to the spherical Darboux frame and investigate the singularities of extrinsic flat great circular surfaces associated with these vector fields. Mathematics Subject Classification. 51H20, 57R45, 58C25. Keywords. The unit 3-sphere, singularity, great circular surfaces, spherical duality.
1. Introduction Touching and slicing a surface are two basic geometric methods to recognize the shape of a surface in the Euclidean 3-space. If we consider the normal slice of the surface with a normal plane at a point, we have a normal curvature of the surface at the point which induces the principal curvatures of the surface. The tangent plane of a surface at a point is determined by the unit normal vector at the point, so that it induces the notion of Gauss maps and the principal curvatures again. These are the methods of slicing and touching for defining the Gaussian curvature and the mean curvature of the surface at a point. Therefore, the curvatures at a point of a surface are obtained using the tangent plane and the normal planes at the point. If we consider a curve on a surface, we have families of tangent planes and rectifying normal planes along the curve, respectively. As the envelopes of these two families of planes, the osculating developable surface and the normal developable surface of the surface along the curve were defined in [3,4,7]. To investigate the singularities of these developable surfaces, some new invariants of the curve on the surface were induced which have interesting geometric properties. The Yang Jiang is supported by the National Natural Science Foundation of China (Grant Nos. 11426157 and 11671070) and Liaoning Natural Science Foundation (Grant No. 20180550233). 0123456789().: V,-vol
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Y. Jiang and S. Izumiya
MJOM
In this paper, we consider the curve on the surface in the unit 3-sphere in the Euclidean 4-space and investigate the properties analogous to those of [3,4,7]. Associated to the curve on the surface in the unit 3-sphere, we have a natural moving frame along the curve which is called a spherical Darboux frame along the curve (cf. Sect. 2). It is the main tool to study the geometric properties of the curves on the surface in the unit 3-sphere. The spherical Darboux frame comprises the position vector of the curve, the unit tangent vector of the curve, the normal vector of the surface along the curve, and the unit tangent vector of the surface which is normal to the curve. The last two vector fields offer important information on the geometric properties of curves on the surface in the unit 3-sphere. We have two special directions in the spherical Darboux frame at each point of the curve whi
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