TITLE: What is the relationship between Grassmann Manifolds with different dimensions?
QUESTION [9 upvotes]: I'm an EE student and I'm just beginning to learn about the Grassmann Manifold.
As is known that the Grassmann Manifold is a space treating each linear subspace with a specific dimension in the vector space $V$ as a single point, for example we can represent the set of all $k$-dimensional linear subspaces $X$ in the $n$-dimensional vector space $V$ as Grassmann Manifold $Gr(k,n)$, and treat each $X \in Gr(k,n)$ as a point in this local Euclidean space.
But I'm wondering that, if we are interested in linear subspaces with different dimensions represented by the Grassmann Manifold, which is $Gr(r,n)$ and $Gr(k,n)$ with $k \neq r$, what is the relationship between them? Is there any theory or book telling this kind of story?
At least I haven't found any material about this question, so I'm hoping anyone who is familiar with this to help me.

REPLY [3 votes]: There are some relationships between the spaces $Gr(1,n), Gr(2,n), \cdots, Gr(n-1,n)$.   The big one (as explained on the Wikipedia page) is that $Gr(j,n)$ and $Gr(n-j,n)$ are diffeomorphic in a canonical way.  The idea is if you have a $j$-dimensional subspace of an $n$-dimensional space, the orthogonal complement is a $n-j$-dimensional subspace.   So you need an inner product to make sense of this, but that's all. 
So when $n=3$, this gives you all the relationships.  When $n$ is even, the orthogonal complement construction gives you a fixed-point free involution of $Gr(n/2,n)$.  
You could ask for other relations but they all have some kind of degeneracy.  For example, if you fix an $n-1$-dimensional subspace of the ambient space, call it $V$.  Then you can intersect a subspace of $Gr(j,n)$ with $V$.  This is "almost" a map from $Gr(j,n)$ to $Gr(j-1,n)$, with the problem being it doesn't take values in $Gr(j-1,n)$ when the vector space in $Gr(j,n)$ is a subspace of $V$.  But if you stare into this construction for a while you develop the idea of "Schubert cells".  This gives you all the main relationships between the grassmannians of various dimensions.  I hope that helps some.