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

Abstract

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 3 1 ) 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, Σ = F −1 (c), 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 3 1 . We develop a systematic approach to the construction of Bishop frames by exploiting the structure of the normal planes induced by the casual character of the curve, present a complete characterization of spherical curves in E 3 1 , 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 3 1 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.

Characterization of manifolds of constant curvature by spherical curves

“…. , n m } along a non-lightlike curve α satisfies equations of motion similar to those found in Lorentz-Minkowski space E 3 1 [6]:…”

“…For E m+1 ν , any hyperquadric Q of radius R with position vector q has ξ = q/R as the unit normal. Then, any curve α : I → Q satisfies ∇ α ′ ξ = α ′ /R, which means that ξ is RM along any curve and, therefore, Q is totally umbilical (see [6], and references therein, for a further investigation of this fact in E 3 1 ).…”

Characterization of Curves that Lie on a Geodesic Sphere or on a Totally Geodesic Hypersurface in a Hyperbolic Space or in a Sphere

“…The basic idea here is that n i rotates only the necessary amount to remain normal to t: in fact, n i is parallel transported along α with respect to the normal connection [14]. Due to their minimal twist, RM frames are of importance in applications, such as in computer graphics and visualization [16,33], sweep surface modeling [3,27,29], and in differential geometry as well [2,11,12,15], just to name a few.…”

On Bishop frame of a pseudo null curve in Minkowski space-time

“…It represents the generalization of the Darboux vector D in E 3 1 . Namely, ifκ 3 = 0 , then timelike curve lies in E 3 1 , so its Darboux vector is given by…”

“…In Minkowski space-time E 4 1 the Bishop frame {T 1 , N 1 , N 2 , N 3 } of a null Cartan curve contains the tangent vector field T 1 of the curve and three vector fields whose derivatives N ′ 1 , N ′ 2 , and N ′ 3 with respect to pseudo-arc are collinear with N 2 [7]. Hence, they make a minimal rotations in the corresponding spaces [6], computer graphics [23], deformation of tubes [21], sweep surface modeling [16], and differential geometry in studying different types of curves (see for example [2,3,15,24]).…”