g_n,k is as described above. An element of g_n,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 g_n,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 Z^p be the subspace of all vectors that are orthogonal to all of the vectors in Z. Geometric intuition for g_3,2 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 g_3,2 but it does for g_4,2.
If X = {<x, y, 0, 0>|x, y ∊ ℝ} ∊ g_4,2
and Y = {<x, y, .1x, .2y>|x, y ∊ ℝ} ∊ g_4,2
then

ⁿ

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 ∊ g_n,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 e_j for a coordinate system for ℝⁿ. For 1≤i≤k, e_i ∊ X and for k<j≤n, e_j is orthogonal to X. Consider a matrix of k(n−k) small reals da_i,j where 1≤i≤k<j≤n. The k vectors f_i = e_i + ∑da_i,je_j (sum k<j≤n) form a nearly orthogonal normal set of vectors that span another nearby element of g_n,k. Each matrix of small da’s define a different element of g_n,k and conversely each element of g_n,k near to X is described by such a matrix. I conclude that g_n,k is a k(n−k) dimensional manifold. ds² = ∑∑da_i,j² defines the same infinitesimal metric as above, I think. I think that the vector space of linear maps from X to X^p forms the tangent space at point X in a Grassmann manifold.

There is a natural isomorphism between g_n,k and g_n,n−k since Z^pp = Z.

Some say that the elements of g_n,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 g_n,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.