- Let $k_1=k_2=1$ be the principal curvatures of a regular surface $S$ at point $p\in S$ and assume that there is a circle $c$ of radius $1/2$ passing through $p.$ Prove that the geodesic curvature of $c$ satisfies $|k_g|\geqslant 1.$
- Let $K=0$ be the Gauss curvature of a regular surface $S$ at point $p\in S$ and assume that there is a circle of radius $1$ and a line passing through $p$ that intersect in right angle. Prove that the mean curvature of $S$ at $p$ satisfies $|H|\leqslant 1/2.$
Attempt.
Since $k_1=k_2$, point $p$ is umbilic and the normal curvature of $S$ at $p$ is constant, $k_n=1$. Since $k^2=k_n^2+k_g^2$ and $k=2$ the curvature of $c$, we get $|k_g|=\sqrt{3}$. So $|k_g|\geqslant 1$. I don't seem to find a flow, but i am wondering why $|k_g|=\sqrt{3}$ was not asked in the initial statement.
Since $K=0$, one of the principal curvatures is zero, let $k_1=0$, so $|H|\leqslant \frac{|k_1|+|k_2|}{2}=\frac{|k_2|}{2}$, so we need to prove that $|k_2|\leqslant 1$. Of course we need to use the fact that a circle and a line intersect in right angle at $p$ - but how does this work?
Thanks in advance.