Grassman manifolds as subsets of Euclidean spaces

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.

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