Grassman manifolds as subsets of Euclidean spaces

TL;DR

This paper explores the geometric structure of Grassman manifolds viewed as subsets of Euclidean spaces, providing explicit formulas for their differential geometry and extending these results to infinite-dimensional Hilbert spaces.

Contribution

It offers explicit formulas for the differential geometry of Grassman manifolds as submanifolds of operator spaces, including extensions to infinite-dimensional cases.

Findings

Derived explicit formulas for the differential geometry of Grassman manifolds.

Extended geometric formulas to infinite-dimensional Hilbert spaces.

Enhanced understanding of 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.