미분기하에서 야코비 정리

미분기하에서 야코비 정리

Jacobi's Theorem in Differential Geometry

Jacobi thm 6.1 p192

가우스 보네 정리의 응용

$\gamma$를 $\mathbb{R}^{3}$의 $\kappa \gt 0$인 정칙곡선이라고 하자. normal spherical image $\sigma$ of $\gamma = N(s)$ of $\gamma$

$\kappa \gt 0$이므로 $N(s)^{\prime} \ne 0$인 정칙곡선이고, $\left\| N(s) \right\| = 1 \implies \sigma : (a,b) \to \mathbb{S}^{2}$

정리

$\gamma$를 단위속력 폐곡선이라 하자. $\sigma$를 단순 곡선이라고 하자. 그러면 $\sigma$는 $\mathbb{S}^{2}$를 면적이 같은 두 영역으로 나누는 곡선이다.

증명

$t$를 $\sigma$의 arc length, $s$를 $\gamma$의 arc length라고 하자. $R$을 $\gamma$로 둘러쌓인 리전이라고 하자.

Claim) $\text{Area}(R)=2\pi$

$\int_{c} k_{g}ds + \text{Area}(R) + \sum \theta_{i} = 2\pi \chi(R)$

$\overline(k_{g}) = \dfac{d}{ds}\left( \arctan \frac{\tau}{\kappa} \right) \dfrac{ds}{dt} = $geodesic curvature of $\sigma$

  1. $\dfrac{d \sigma}{d t} = \dfrac{d N}{ds}{ds}{dt} = (-\kappa T + \tau B)\dfrac{ds}{dt}$

$1 = \left\| \dfrac{d\sigma}{dt} \right\| = \left\| (-\kappa T + \tau B)\dfrac{ds}{dt} \right\| = \left| \dfrac{ds}{dt} \right| \sqrt{\kappa^{2} + \tau^{2}} \implies \dfrac{ds}{dt} = \dfrac{1}{\sqrt{\kappa^{2} + \tau^{2}}}$

$$ \dfrac{d^{2}\sigma}{d t^{2}} = \dfrac{d^{2} N}{d t^{2}} = \dfrac{d}{dt}((-\kappa T + \tau B)\dfrac{ds}{dt}) = (-\kappa T + \tau B)\dfrac{d^{2}s}{dt^{2}} + \dfrac{d}{ds}(-\kappa T + \tau B)(\dfrac{ds}{dt})^{2} = -\kappa ^{\prime} T - \kappa T^{\prime} + \tau^{\prime} B + \tau B^{\prime} = -\kappa^{\prime} T - \kappa(\kappa N) + \tau^{\prime}B + \tau(-\tau N) = (-\kappa T + \tau B)\dfrac{d^{2}s}{dt^{2}} + (-\kappa^{\prime} T -(\kappa^{2}+\tau^{2})N + \tau^{\prime} B)(\dfrac{ds}{dt})^{2} $$

intrinsic normal $S_{\sigma} = N \times \dfrac{d N}{d t}$

$\overline(k_{g}) = \left\langle N \times \dfrac{dN}{dt}, \dfrac{d^{2}N}{dt} \right\rangle = \left\langle N, \dfrac{dN}{dt} \times \dfrac{d^{2}N}{dt} \right\rangle = \left\langle N, (-\kappa T + \tau B)\dfrac{ds}{dt} \times (-\kappa^{\prime} T - (\kappa^{2}+\tau^{2})N + \tau^{\prime}B)(\dfrac{ds}{dt})^{2} \right\rangle = (\dfrac{ds}{dt})^{3} \left\langle N, \kappa(\kappa^{2}+\tau^{2})T \times N - \kappa \tau^{\prime} T\times B - \tau \kappa^{\prime}B\times T - \tau(\kappa^{2}+\tau^{2})B\times N \right\rangle$

$\implies \overline{k_{g}} = (\dfrac{ds}{dt})^{3} \left\langle N,(\kappa \tau^{\prime} - \tau \kappa^{\prime})N \right\rangle = (\dfrac{ds}{dt})^{3} (\kappa\tau^{\prime} - \tau \kappa^{\prime}) = \dfrac{ds}{dt} \dfrac{d}{ds} \arctan (\dfrac{\tau}{\kappa})$

$$ \int_{\sigma} \overline{k_{g}}dt = \int_{\sigma} \dfrac{ds}{dt} \dfrac{d}{ds} \arctan (\dfrac{\tau}{\kappa}) dt = \int_{\sigma} \dfrac{d}{ds} \arctan (\dfrac{\tau}{\kappa}) ds = 0 (by closed) $$

$$ \text{Area}(R) = \iint_{R}dA = \iint_{R}K dA = 2\pi\chi(R)\int_{\sigma}\overline{k_{g}}dt = 2\pi - \int_{\sigma}\overline{k_{g}}dt = 2\pi $$

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