All Questions

Let $S$ be a smooth surface embedded in $\mathbb{R}^3$. When are (some of) the principal lines of curvature geodesics on $S$? Perhaps on the ellipsoid below, the (blue) central cycle, a max principal ...

An arc is a set homeomorphic to the unit interval $[0,1]$; an arc in $\mathbb{R}^3$ is planar if it is contained in some plane. The following questions are motivated by Anton Petrunin's Disc bounded ...

This is higher dimension conjecture of Problem 3845 in Crux Mathematicorum and Theorem 2 in here: PS: This figure is very nice, this is also generalization of Brianchon’s theorem, The Pascal theorem, ...

Let $\Sigma$ be a sphere topologically embedded into $\mathbb{R}^3$. Is it always possible to find a disc $\Delta\subset\Sigma$ which is bounded by a plane curve? It is easy to find an open disc ...

Let $u$ and $v$ be two points on a surface (I guess, a Riemann surface) $\Sigma$ such that there is a unique geodesic between $u$ and $v$ on $\Sigma$. Now let $l$ be an arbitrary line that passes ...

Let consider a surface $z=1/(xy)$ and a sphere defined by $(x-1.5)^2+(y-1.5)^2+(z-1.5)^2=3/4$. The sphere touches the surface at (1,1,1). Is it possible to prove that point (1,1,1) is the only ...