Generalization of Vogt's Theorem for curves in higher dimension

2012-10-22T07:27:36Z

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 $\gamma : [0, l] \rightarrow \mathbb{R}^3$, 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.

Answer by Sunayana Ghosh for Generalization of Vogt's Theorem for curves in higher dimension

2013-01-18T15:54:29Z

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.

Generalization of Vogt's Theorem for curves in higher dimension - MathOverflow

Generalization of Vogt's Theorem for curves in higher dimension

Answer by Sunayana Ghosh for Generalization of Vogt's Theorem for curves in higher dimension