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Статья опубликована в журнале из перечня ВАК
Статья опубликована в журнале из списка Web of Science и/или Scopus
Дата последнего поиска статьи во внешних источниках: 9 декабря 2020 г.
Аннотация:Intersections of cones of index zero with spheres are investigated. Fields of the corresponding minimal manifolds are found. In particular, the cone K={x02+x12+x22+x32} is considered. Its intersection with the sphere S3=∑i=03xi2 is often called the Clifford torus T, because Clifford was the first to notice that the metric of this torus as a submanifold of S3 with the metric induced from S3 is Euclidian. In addition, the torus T considered as a submanifold of S3 is a minimal surface. Similarly, it is possible to consider the cone K={∑i=03x02=∑i=47xi2}, often called the Simons cone because he proved that K specifies a single-valued nonsmooth globally defined minimal surface in ℝ8 which is not a plane. It appears that the intersection of K with the sphere S7, like the Clifford torus, is a minimal submanifold of S7. These facts are proved by using the technique of quaternions and the Cayley algebra.