Grassman manifolds as subsets of Euclidean spaces

Abstract

We consider the Grassman manifold $G(E)$ as the subset of all orthogonal projections of a given Euclidean space $E$ and obtain some explicit formulas concerning the differential geometry of $G(E)$ as a submanifold of $L(E,E)$ endowed with the Hilbert-Schmidt inner product. Most of these formulas can be naturally extended to the infinite dimensional Hilbert space case.

Latex-reformatted version of the conference article "Grassman manifolds as subsets of Euclidean Spaces", in Differential Geometry, Proc. 5th Int. Colloq., Santiago de Compostela, Spain, 1984, Research Notes in Mathematics, 131, 85-102, Pitman Adv. Pub. Prog., Boston-London-Melbourne (1985); scans of original paper attached as figures