After seeing this brief description of a Grassmann Manifold I thought about it for a while and decided to write some notes before searching the web for more information. This is thus probably a poor place to start learning about Grassmann Manifolds. This note is mostly chronological and records my gropings. There remain serious bugs. The coördinate free scheme comes from intuition for g3,2 and algebraic examples from g4,2.
This is an unfruitful search for a numerical relation between subspaces of a Euclidean space. I now think that the relation between two subspaces B and C of an n dimensional vector space with positive definite quadratic form is entirely captured by the shape of the following body: Take the unit sphere in B (or C), orthogonally project it into the other sub-space, and then orthogonally project it back. The result will be an ellipsoid whose shape (in turn captured by the lengths of its axes), captures the metric relationship between the two subspaces.
The eigenvalues of these double projections are all between 0 and 1 inclusive. If such eigenvalues are all 0 then the spaces are orthogonal. If they are all 1 then one space is included in the other. These eigenvalues (or axis lengths) are cosines of the angles between the two spaces in a sense defined hereby.
By ‘capturing the numerical relation’ I mean that for two configurations yielding congruent doubly projected bodies, a rigid rotation of the embedding space can bring one configuration to the other; the first pair of subspaces is congruent to the second pair.