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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.

Helmholz
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    Just curious: Why are you learning Grassman Manifolds? I am an EE student too, but when I want to learn about this type of stuff I have to teach myself outside of class (it isn't in the flowchart). As for your question, I don't think there are any obvious connections between the different dimensions, but maybe I haven't learned enough yet to know the right answer. – Ryan Sep 16 '13 at 04:18
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    Well, to be honest, I'm just doing some research about Signal Processing where subspace-modeled signals (the signal that comes from a specific linear subspace)is of my interest. So it just comes to me that the mathematical model of Grassmann Manifold can be a new tool to solve my signal processing problems. And of course this stuff is like some bad dream to me, but I have to read those language from Mars all by myself... Anyway, thank you for your comment:), and may there be chances that we can discuss further about this in the future~~ – Helmholz Sep 16 '13 at 05:20
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    You can maybe look up flag varieties; they are generalizations of $Gr$, where you have several linear subspaces – user8268 Sep 16 '13 at 06:35

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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.

Ryan Budney
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