What is the relationship between Grassmann Manifolds with different dimensions? 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.
 A: 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. 
