Moving frames and the characterization of curves that lie on a surface

In this work we are interested in the characterization of curves that belong to a given surface. To the best of our knowledge, there is no known general solution to this problem. Indeed, a solution is only available for a few examples: planes, spheres, or cylinders. Generally, the characterization of such curves, both in Euclidean (E^3) and in Lorentz–Minkowski (E_1^3) spaces, involves an ODE relating curvature and torsion. However, by equipping a curve with a relatively parallel moving frame, Bishop was able to characterize spherical curves in E^3 through a linear equation relating the coefficients which dictate the frame motion. Here we apply these ideas to surfaces that are implicitly defined by a smooth function, S = F^{-1}, by reinterpreting the problem in the context of the metric given by the Hessian of F, which is not always positive definite. So, we are naturally led to the study of curves in E_1^3. We develop a systematic approach to the construction of Bishop frames by exploiting the structure of the normal planes induced by the causal character of the curve, present a complete characterization of spherical curves in E_1^3 , and apply it to characterize curves that belong to a non-degenerate Euclidean quadric. We also interpret the casual character that a curve may assume when we pass from E^3 to E_1^3 and finally establish a criterion for a curve
to lie on a level surface of a smooth function, which reduces to a linear equation when the Hessian is constant.