gn,k is as described above. An element of gn,k is a k dimensional subspace of the vector space ℝn. It seems clear when two subspaces are near, at least if we as assume the Euclidian metric, d(x, y) = |x−y|, normally ascribed to ℝn. We have a good concept for two unit vectors being close. We consider distance between two unit vectors of the respective subspaces X and Y. Let X' be the unit vectors of X and ditto Y'. Neither inf{d(x, y)|x∊X' ⋀ y∊Y'} nor sup{d(x, y)|x∊X' ⋀ y∊Y'} will do. If two elements X and Y of gn,k have an intersection with more than zero dimensions, which is certain if 2k>n, then a unit vector in the intersection belongs to both X' and Y'. If z∊X'∩Y' then d(z, z) = 0 and d(z, −z) = 2. Thus inf{} = 0 and sup{} = 2 neither of which bears on the distance from X to Y. We must be more careful about what pairs of vectors to select from X and Y. If Z is a subspace let Zp be the subspace of all vectors that are orthogonal to all of the vectors in Z. Geometric intuition for g3,2 suggests that we choose x from (X∩Y)p∩X and y from (X∩Y)p∩Y. The angle between x and y does not depend on this choice for g3,2 but it does for g4,2.
If X = {<x, y, 0, 0>|x, y ∊ ℝ} ∊ g4,2
and Y = {<x, y, .1x, .2y>|x, y ∊ ℝ} ∊ g4,2
then
X and Y are not orthogonal. (X∩Y) = {0} and (X∩Y)p = ℝn. A unit element of X is closer to some unit elements of Y than to others. {d(x, y)|x∊X ⋀ y∊Y ⋀ |x| = |y| = 1} ≃ [.1, 2].
Perhaps the lower bound is the appropriate estimate:
d(X, Y) = inf {d(x, y)|x∊(X∩Y)p∩X ⋀ y∊(X∩Y)p∩Y ⋀ |x| = |y| = 1} should serve only as an infinitesimal metric. It may then be used to define geodesics, and hopefully a Riemannian metric. (I am pretty sure it is at least a Finsler space.)

For some purposes the image of the unit sphere projected into the other space is a simpler construct. It is a central ellipsoid which is a subset of the local unit sphere. It conveys the same information. It has no interesting eigen vectors, however.

The above idea is coördinate free. Next we develop of another metric based on coördinates which I think is equivalent. For an element X ∊ gn,k choose k mutually orthogonal unit vectors in X. Now choose n−k more unit vectors in ℝn that are mutually orthogonal and also orthogonal to X. We have thus a set of n orthogonal basis elements ej for a coordinate system for ℝn. For 1≤i≤k, ei ∊ X and for k<j≤n, ej is orthogonal to X. Consider a matrix of k(n−k) small reals dai,j where 1≤i≤k<j≤n. The k vectors fi = ei + ∑dai,jej (sum k<j≤n) form a nearly orthogonal normal set of vectors that span another nearby element of gn,k. Each matrix of small da’s define a different element of gn,k and conversely each element of gn,k near to X is described by such a matrix. I conclude that gn,k is a k(n−k) dimensional manifold. ds2 = ∑∑dai,j2 defines the same infinitesimal metric as above, I think. I think that the vector space of linear maps from X to Xp forms the tangent space at point X in a Grassmann manifold.

There is a natural isomorphism between gn,k and gn,n−k since Zpp = Z.

Some say that the elements of gn,k are oriented. Such a manifold has two elements where the vanilla sort has just one by identifying antipodal points of the manifold.

### Questions

What relations have two elements of gn,k aside from the distance between them? (there are some)
What is the group of a Grassmann manifold? (depends on previous question)
Does the metric from geodesics correspond to d(X, Y) taken non-infinitesimally in a simple functional manner?

Now I think the above two metrics are not equivalent.

### A Conjecture

Consider two k dimensional subspaces, X and Y of ℝn. Let P be the projection from X to Y (P : X → Y) and Q the projection from Y to X (Q : Y → X). The eigenvalues of PQ ⊂ [0, 1]. Counting multiplicities, the eigenvalues characterizes the relation between X and Y up to congruence. If subspaces U and V produce the same sorted set of eigenvalues then some orthonormal transformation of ℝn will transform U to X and Y to V. The set of angles cos−1(√e) for each of the eigenvalues e could be defined as the angles between two subspaces.