Euclidean and timelike bishop spherical curves and their characterizations

One of the most important topics of differential geometry is the study of space curves. Frenet frame consisting of tangential, principle normal and binormal vectors is used to characterize curves in 3-dimensional Euclidean space. This is the best orthonormal frame to study space curves. In addition, the curvature and torsion of a curve informs about the local behavior of this curve in space. The curvature of a curve indicates the tangential direct deviation of this curve, the torsion of the defined curve also measures the amount of deviation from the osculating plane determined by the tangent and normal vectors of this curve. In general, a space curve can be characterized by a differential equation including curvature and torsion. Fort this reason, a curve must be able to be differentiated at least up to the third order in order to fully examine it. The curvature of a curve at some points can be vanished which means that the second derivative can be zero. In this case, an alternative to the Frenet frame and associated with it that is called Bishop frame can be created for better examination of the curve. Without changing the tangent vector on the Frenet frame, another original frame called the Bishop frame or parallel translational frame is obtained by rotating the principal normal and binormal vectors at an angle. Accordingly, the vector remains in the Bishop frame, and any two element present in the plane perpendicular to this vector is selected. The derivatives of these two vectors are linked only to the vector. Therefore, there is a relationship between the first and second curvatures of this curve according to Bishop frame and the curvature and torsion of this curve. Recently, many different characterizations of space curves are given using the Bishop frame. One of the most important space curves is the spherical curves lying on the sphere. In this study, integral characterizations of a spherical curve according to Frenet frame are investigated. Particularly, necessary and sufficient condition for a curve to be a spherical curve has been demonstrated by using the first and second curvatures according to the Bishop frame. A third-order differential equation is used to characterize a spherical curve using the Bishop frame . Also, a third-order differential equation is used to characterize a spherical curve using timelike Bishop frame .