Narrow Results

Let $q$ be a prime of the form $4k+3$. Then for any three mutually orthogonal great circles of the sphere $x^2+y^2+z^2=q$ lying on rational planes, at least one of these circles does not contain any rational points.

We characterize Clifford minimal hypersurfaces S^r(nc/r) × S^n-r(nc/(n - r)) with 1 ≦ r ≦ n - 1 in a sphere Sⁿ⁺¹(c) of constant sectional curvature c by observing their geodesics from this ambient sphere.