For any Sequence of Integers , there is a flag manifold of type (, ..., ) which is the collection of ordered pairs of vector Subspaces of (, ..., ) with and a Subspace of . There are also Complex flag manifolds with Complex subspaces of instead of Real Subspaces of a Real -space. These flag manifolds admit the structure of Manifolds in a natural way and are used in the theory of Lie Groups.
See also Grassmann Manifold
Lu, J.-H. and Weinstein, A. ``Poisson Lie Groups, Dressing Transformations, and the Bruhat Decomposition.''
J. Diff. Geom. 31, 501-526, 1990.