The faces of the Grassmannian of 3-planes in \({\mathbb{R}}^ 7\) (calibrated geometries on \({\mathbb{R}}^ 7)\)
A classification is given of the faces of the Grassmannian of oriented 3- planes in \({\mathbb{R}}^ 7\), and hence of the calibrated geometries of 3-dimensional area-minimizing surfaces in \({\mathbb{R}}^ 7\). There are five discrete types of faces and five infinite families of types. New phenomena include faces which are not totally geodesic in the Grassmannian: nonround \(S^ 1\)'s, \(S^ 2\)'s, and \(S^ 3\)'s. Classification of calibrated geometries in \({\mathbb{R}}^ 6\), the first nonclassical case, was completed in 1982 by \textit{J. Dadok} and \textit{R. Harvey} [Duke Math. J. 50, 1231-1243 (1983; Zbl 0535.49030)] and \textit{F. Morgan} [Lin. Alg. Appl. 66, 1-28 (1985)]. Results are forthcoming on \({\mathbb{R}}^ 8\) by Dadok, Harvey, and Morgan and on Grassmannians by Gluck, Morgan, and Ziller.
scientific article; zbMATH DE number 3886696