On the energy and pseudoangle of Frenet vector fields in $ R_v^n $

Abstract

In this paper, we compute the energy of a Frenet vector field and the pseudo-angle between Frenet vectors for a given non-null curve C in semi-Euclidean space of signature (n; ). It is shown that the energy and pseudo-angle can be expressed in terms of the curvature functions of C. ˛AE˜aeoÂIO ÂIÂ﷿ai˛ ‚ÂŒUO﷿IOaO OOoˇ ‘﷿ÂI U‡ Oae‚‰OŒUU IiE ‚ÂŒUO﷿‡I ‘﷿ÂI ‰oˇ A‡‰‡IO¤ IÂIUo¸O‚O¤ Œ﷿‚O¤ C U I‡Oi‚‚Œoi‰O‚OIU O﷿OaeUO﷿i aeaI‡UU﷿ (n; ). ˇOŒ‡A‡IO, oO ÂIÂ﷿aiˇ U‡ Oae‚‰OŒUU IOEUU¸ AEUU ‚﷿‡EÂIi ˜Â﷿ÂA UUIŒˆi¤ Œ﷿‚I C. 1. Introduction. It is well known that the studies on the energy of a unit vector field on a compact oriented m-dimensional Riemannian manifold M basically consider the equality M = S 2n+1 (see [1 ‐ 3]). Let C be a curve with a pair (I; ) of parametric unit speed in R n . Let us take an initial point a 2 I and the Frenet frames fV1( (a));:::;Vr( (a))g andfV1( (s));:::;Vr( (s))g at the points (a) and (s), respectively. In [4], we calculated the energy of a Frenet vector field and the angle between each vector Vi( (a)) and Vi( (s)) where 1 i r. Further, we observed that the energy and angle may be expressed in terms of the curvature functions of the given curve C. In this paper, we consider the Frenet frame at the point (a) for a given non-null curve C in semi-Euclidean space R n = R n ; X