Minimal Submanifolds of Spheres and Cones

Intersections of cones of index zero with spheres are investigated. Fields of the corresponding minimal manifolds are found. In particular, the cone $$\mathbb{K} = \left\{ {x_0^2 + x_1^2 + x_2^2 + x_3^2} \right\}$$ is considered. Its intersection with the sphere $$\mathbb{S}^{3}=\sum\nolimits_{i=0}^{3} x_{i}^{2}$$ is often called the Clifford torus $$\mathbb {T}$$, because Clifford was the first to notice that the metric of this torus as a submanifold of $$\mathbb {S}^3$$ with the metric induced from $$\mathbb {S}^3$$ is Euclidian. In addition, the torus $$\mathbb {T}$$ considered as a submanifold of $$\mathbb {S}^3$$ is a minimal surface. Similarly, it is possible to consider the cone $${\mathcal K} = \{ \sum\nolimits_{i = 0}^3x_0^2 = \sum _{i = 4}^7x_i^2\} $$, often called the Simons cone because he proved that $${\mathcal K}$$ specifies a single-valued nonsmooth globally defined minimal surface in ℝ8 which is not a plane. It appears that the intersection of $${\mathcal K}$$ with the sphere $$\mathbb{S}^7$$, like the Clifford torus, is a minimal submanifold of $$\mathbb{S}^7$$. These facts are proved by using the technique of quaternions and the Cayley algebra.