The Vogt's theorem for plane curves states that if A and B are endpoints of a spiral arc, the curvature nondecreasing from A to B. The angle $\beta$ of the tangent to the arc at B with the chord AB is not less than the angle $\alpha$ of the tangent at A with AB. $\alpha = \beta$ only if the curvature is constant.
Does anyone know of a result which extend this theorem to space curves or curves in higher dimension. I have the following conjecture for space curves: Given a regular curve in space $\mathbb{R}^3$ with \gamma : [0, l] \rightarrow \mathbb{R}^3$, parametrized by arc-length $s$, let $\kappa$ and $\tau$ denote the Euclidean curvature nondecreasing between endpoints and torsion respectively. Let us assume that $\kappa$ is non-decreasing and $\tau$ is non-decreasing. Let $A = \gamma(0)$ and $B = \gamma(l)$ and torsion non-zero everywhere. The angle let $\beta$ of \alpha$ be the angle between the tangent plane to the arc at B with $\gamma(0)$ and the chord AB is not less than $AB$ and let $\beta$ be the angle $\alpha$ of between the tangent plane at A with AB. $\gamma(l)$ and the chord $AB$. We claim that $\alpha = \leq \beta$ and equality holds only if the curvature and torsion are constant,i.e., when the curve $\gamma$ is a circular helix.
I would be quite glad to receive any tips or counter examples and whether anybody already knows a solution to this problem. I also believe that this result can be extended to curves in $\mathbb{R}^n$.
Thanks.
