Generalization of Vogt’s Theorem for curves in higher dimension

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1. 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.

2. 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 $\gamma : [0, l] \rightarrow \mathbb{R}^3$, parametrized by arc-length $s$, let $\kappa$ and $\tau$ denote the Euclidean curvature 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 let $\alpha$ be the angle between the tangent plane at $\gamma(0)$ and the chord $AB$ and let $\beta$ be the angle between the tangent plane at $\gamma(l)$ and the chord $AB$. We claim that $\alpha \leq \beta$ and equality holds only if $\gamma$ is a circular helix.

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1 Answer

This is an attempt at an (as yet incomplete) proof of the above claim. I would be happy to receive any corrections and/or comments.

Let us denote the curve $\gamma$ by the following parametrization $\gamma(s) = (x_1(s), x_2(s), x_3(s))$. Without loss of generality let us assume that $A = \gamma(0) = (0, 0, 0)$ and $B = \gamma(l) = (x_1(l), 0, 0)$.

Let $\theta(s)$ denote the angle between tangent plane at $\gamma(s)$ and the chord $AB$, thus we have that: $\sin \theta(s) = \langle B(s), (1,0,0) \rangle$. Using Frenet-Serret formulae where :

$T'(s) = \kappa(s) N(s)$ and $N'(s) = -\kappa(s) T(s) -\tau(s) B(s)$

we have that : $\sin \theta(s) = \langle B(s), (1,0,0) \rangle = \frac{1}{\kappa(s)} (\gamma'(s) \times \gamma''(s)) = \frac{1}{\kappa(s)} (x_2'(s) x_3''(s) - x_3'(s) x_2''(s))$.

Claim : $\alpha \leq \beta$ i.e., it is enough to prove that $\int_{\theta(0)}^{\theta(l)} \sin \theta d\theta \geq 0$.

From the equation for $\sin \theta(s)$ we obtain that $d\theta(s) = \frac{\kappa(s) f'(s) - f(s) \kappa'(s)}{\kappa(s)\sqrt{\kappa(s)^2-f(s)^2}}$, where $f(s) := x_2'(s) x_3''(s)-x_2''(s) x_3'(s) = \kappa(s) \langle B(s), e_1 \rangle$ and $e_1:= (1,0,0)$.

On further simplification using Frenet-Serret formulae we get:

$\int_{\theta(0)}^{\theta(l)} \sin \theta(s) d\theta(s) = \int_0^l \frac{\tau(s)}{\kappa(s)} \frac{\langle B(s), e_1 \rangle \langle N(s), e_1 \rangle}{\sqrt{1-\langle B(s), e_1\rangle ^2}} ds$.

From here it is not clear to me that product of the integrand is always positive, or using integration by parts the integrand is always positive.

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