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I've studied differential geometry and get this question.

I'd like to verify following statement.

curve of constant curvature on unit sphere is planar curve

I've struggled with Frenet-Serret frame, differentiating, differentiating, differentiating, .....

BUT I didn't get something yet..

Could you give me some hint, please?

$$ $$ Ah!! FIRST OF ALL,

I'd like to know whether the statement is true or false.

Thanks in advance.

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You want to show that the torsion is zero. Note that If $\mathbf{r}$ is the unit radius vector to the curve then $$\mathbf{t}=\frac{d \mathbf{r}}{ds}$$ Now $\mathbf{t} \cdot \mathbf{r}=0$ so on differentiating we get

$$\frac{d \mathbf{t}}{ds}\mathbf{r}+\mathbf{t}\frac{d \mathbf{r}}{ds}=0$$ or $$\kappa \mathbf{n}\cdot \mathbf{r}+ \mathbf{t}\cdot \mathbf{t}=0$$ (Note that $\kappa\neq 0$, since otherwise we have a straight line.) So $$\mathbf{n}\cdot \mathbf{r}=-\frac{1}{\kappa}$$ differentiating this and using that $\kappa$ is constant we get

$$(-\kappa \mathbf{t}+\tau \mathbf{b})\cdot \mathbf{r}+\mathbf{n}\cdot \mathbf{t}=0$$ which simplifies to $$\tau \mathbf{b}\cdot \mathbf{r}=0$$ If $\tau=0$ we are done so assume

$$\mathbf{b}\cdot \mathbf{r}=0$$

and differentiate which gives

$$-\tau \mathbf{n}\cdot \mathbf{r}+\mathbf{b}\cdot \mathbf{t}=0$$

$$-\tau \mathbf{n}\cdot \mathbf{r}=0$$ But $\mathbf{n}\cdot \mathbf{r}=-\frac{1}{\kappa}$ so again $\tau=0$.

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  • $\begingroup$ so clear... THANK you very much. $\endgroup$ – user143993 Jul 4 '14 at 16:07
  • $\begingroup$ It was a pleasure. $\endgroup$ – Rene Schipperus Jul 4 '14 at 16:09

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