Constant Scalar Curvature of Three Dimensional Surfaces Obtained by the Equiform Motion of a Sphere

The number s is called the scaling factor. An equiform motion is defined if the parameters of (1.1), including s, are given as functions of a time parameter t. Then a smooth one-parameter equiform motion moves a point x via x(t) = s(t)A(t)x(t) + d(t). The kinematic corresponding to this transformation group is called equiform kinematic. See [2, 4]. Under the assumption of the constancy of the scalar curvature, kinematic surfaces obtained by the motion of a circle have been obtained in [1]. In a similar context, one can consider hypersurfaces in space forms generated by one-parameter family of spheres and having constant curvature: [3, 5, 6, 7]. In this paper we consider the equiform motions of a sphere k0 in E . The point paths of the sphere generate a 3-dimensional surface, containing the positions of the starting sphere k0. Corresponding author. E-mail address: [email protected] (A. T. Ali).