The isoperimetric inequality and the total absolute curvature of closed curves in spheres (Q1197350)

Continuing his research published in Manuscr. Math. 57, 101-108 (1986; Zbl 0589.53001) the author proves the inequality \(4(1-(L/2\pi)^ 2)\leq \text{tac}\), where \(L\) is the length and tac is the total absolute (geodesic) curvature of a closed curve in a unit sphere. The equality just holds for one period of a geodesic circle. Moreover, for closed curves in the 2-sphere he obtains a lower bound for the tac depending on the maximal winding number and the length of the curve.