Problem $3\mathrm{.}6.[3$ pts$]$ Let $$c$()$ : $$ R$3$ be a $2$ curve on the sphere of radius $$ at the origin.

$2$

Mahan Moazzeni

Prove that the curvature of $$c$()$ is bounded from below by the reciprocal of the radius of the sphere, i$.$e$.,$

$()1$

$$ for every $$ in $.$

And also show that equality $()=1$

$$ holds if and only if the shape of the curve $$c$()$ is a great circle $($the

intersection of the sphere with a plane through the origin$).$