i have a curve $\alpha:I\rightarrow \mathbb{R}^3$ such that his curvature $k$ is constantand $\alpha$ is entirely contained in a sphere, i must prove that this curve is a cirlce.

My try:

I need to prove that $\alpha$ has zero torsion, i supose the sphere with center at origin and radius $r>0$, so i got $|T^{'}(s)|=k$ for all $s$ where $T$ is the tangent line of $\alpha$ and $|\alpha(s)|^2=r^2$ for all $s$, i wanted to prove that the binormal $B$ of $\alpha$ is contants, but i derivatite these two equation a lot and i couln't conclude anything, anyone can help?

  • $\begingroup$ what do you know about $\alpha \cdot \alpha \; ?$ It appears you are switching between $\alpha$ and $\gamma$ $\endgroup$ – Will Jagy Jan 21 '18 at 0:02
  • $\begingroup$ that is equal to the radius of the sphere. P.S. sorry, i edited $\endgroup$ – Eduardo Silva Jan 21 '18 at 0:04
  • $\begingroup$ no. That is the square root. I want you to tell me about the derivative of $\alpha \cdot \alpha$ by the variable $s$ $\endgroup$ – Will Jagy Jan 21 '18 at 0:05
  • $\begingroup$ thats right, but i tried this as you can see in my question, so how this is going to work? the derivatives will give me that $\alpha$ and its tangents are perpendicular $\endgroup$ – Eduardo Silva Jan 21 '18 at 0:07
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    $\begingroup$ How do you write $\alpha$ in terms of $T,N,B \; ?$ $\endgroup$ – Will Jagy Jan 21 '18 at 0:09

thanks to the coments of @willjagy i could conclude my demonstrations, first i find by derivating $|\alpha(s)|^2$ that $<T,\alpha>=0$, derivating again and using Frenet Formulas i get

$k<N,\alpha>+1=0$ and one more time:

$k^{'}<N,\alpha>-k\tau<B,\alpha>=0$, where $\tau$ is the torsion of the curve, so, as $k$ is contant and positive:

$\tau<B,\alpha>=0$ $ \forall s\in I$,

and then at any cases $\tau=0$ or $<B,\alpha>=0$ i can conclude the $\alpha$ is a circle, for the first i get $\alpha$ in a great circle of the sphere and for the second an arbitrary cirlce with radius less or equal than $r$.

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    $\begingroup$ not quite. If we introduce two functions of $s,$ call them $p$ and $q,$ and say $\alpha = p N + q B,$ differentiate and collect the coefficients of $T,$ then of $N,$ then of $B,$ the conclusions are, given constant $k,$ that both $p$ and $q$ are constant, $p$ is nonzero, so $\tau$ is zero. $\endgroup$ – Will Jagy Jan 21 '18 at 2:26
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    $\begingroup$ If you think geometrically, two things are wrong here. First, $N$ points inward and $\alpha$ points outward. Second, the only time this equation can hold is for a great circle (the intersection of the sphere with a plane through the origin — the center). $\endgroup$ – Ted Shifrin Jan 21 '18 at 5:04
  • $\begingroup$ By your coments i tried again and reformulate my argumentation, any gap now? $\endgroup$ – Eduardo Silva Jan 21 '18 at 15:09

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