I wonder what's the special about grassmannian space? Why we want to use this space?

On wikipedia, it says:

"By giving a collection of subspaces of some vector space a topological structure, it is possible to talk about a continuous choice of subspace or open and closed collections of subspaces; by giving them the structure of a differential manifold one can talk about smooth choices of subspace. " As a student has few knowledge in differential geometry, I found it hard to catch the meaning of it.

Could anyone gives me some more intuition to understand it?



Here are two examples:

First, suppose you have a region in $\mathbb R^n$ and you wish to study the $k$-dimensional area of its projections onto all possible $k$-dimensional subspaces of $\mathbb R^n$. This defines a function on the Grassmannian $G(k,n)$ of $k$-planes in $\mathbb R^n$. You might even want to integrate that function and get the average shadow of the region.

Second, perhaps most fundamental, suppose $M\subset\mathbb R^n$ is a smooth submanifold. We can define the Gauss map $M\to G(k,n)$, which assigns to each point $p$ of $M$ its tangent plane $T_pM$. Properties of this mapping are fundamental in differential geometry, and deep theorems can be proved by studying the universal setting of the "tautological" vector bundle $\xi\to G(k,n)$.

  • $\begingroup$ Thanks for the example, that's very helpful ! So $G(k,n)$ is a collection of these projections, and it is discontinuous set because you have one projection on one set of $k$ dimension you picked? Also, could you please explain a little bit more on why we care about the map to the tangent plane ? That's really helpful for me. $\endgroup$
    – Jack2019
    Jul 11 '13 at 15:55
  • $\begingroup$ Not quite. For each $k$-dimensional subspace $V\subset\mathbb R^n$, I'm defining $f(V)$ to be the area of the projection of the given region onto $V$. So this defines a function $f\colon G(k,n)\to\mathbb R$. ... Generally, in topology and geometry we want to study tangent planes of (higher-dimensional) surfaces, called manifolds, and how they vary from point to point. Curvature of a curve is a measure of how quickly its tangent line is turning in space; curvature of a surface is a measure of how quickly its tangent plane is turning in space. Etc. $\endgroup$ Jul 11 '13 at 16:48

Consider $G(1,K^3)$, where $K$ is a field. This takes the collection of lines through the origin in the field's 3D space and turns them into points, but it also preserves incidence. So it takes planes containing those lines and turns them into a line which contains all the original lines as its points. Imagine the set of lines through the origin of a plane. While there's one for each slope in the plane, the plane also has a vertical sloping line. Hence, the line each plane is turned into has one more point over lines in a $K$-vector space. This is a projective line the plane has turned into. But moreover, for each plane through the origin in the original space, there's a line of intersection between them, hence there's a point of intersection in the lines of the Grassmannian. This Grassmannian is the projective plane for that field.

The connection to general Grassmannians is that, when your eye receives images, it's classifying rays of light that run through your eye, and treating each class, or angle light can hit it from, as a point. And in general, if something can only detect a geometric object up to a class of subspace it belongs to, a different geometry will arise in the detector than the reality it's modelling.

From here several motivations can lead one to take interest in Grassmannians that take higher subspaces as equivalent at a time.

If one wishes to see 4D objects, one usually takes the projection into 3D first, then another projection from 3D to 2D. Can you instead classify planes of 4D as a whole, and will that give you a different means of visualization, perhaps more natural? Perhaps one just wants to find interesting interactions between different dimensions of space. Grassmannians are a way of simulating having senses with a different dimensional relationship to the space around them.

Frequently, one investigates the ways of classifying a type of object completely. This is to help see what phenomena are possible in a field of study. But instead of relying on a simple taxonomy based on observed properties, one might wonder if there's actually a space the objects under study naturally sit in, a classifying space for a set of invariants.

And by space you might be talking coordinate system, a way of moving between different classes of objects (as one can move between quadratic polynomials or the values for their roots by scaling their coefficients), and you might be talking topology, a connection or common region of variation between some of the objects and perhaps not others. Or perhaps the space has literal holes where phenomena aren't observed, where some limits can be seen for properties like the curvature or irregularity of the objects. That allows one to speak of these constraints as variational extremes, which changes one's relationship to results on what is and isn't possible by collecting many just-so answers into one physical boundary. And perhaps one can even take the direction the constraints are moving in when they reach their limits and derive the direction the object would have to be generalized in to get objects which take on those forbidden properties.

So in this light, Grassmannians are the simpler solutions to the general problem of constructing classifying spaces and moduli spaces in particular, and indeed a lot of what people adopt as constructions for these types of spaces are motivated by taking how Grassmannians behave as principals for how classifying spaces should behave.

  • $\begingroup$ Thank you very much for your help. It's very motivating. $\endgroup$
    – Jack2019
    Jul 11 '13 at 16:02
  • $\begingroup$ @SoManyProb_for_a_broken_heart. No problem. $\endgroup$
    – Loki Clock
    Jul 12 '13 at 6:20

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