A natural embedding of the total space of tautological bundle over $G(2,n)$ in $G(2,n+1)$

Question

I learned from the following post that the total space of the tautological line bundle over $\mathbb{R}P^{n}$ is diffeomorphic to $\mathbb{R}P^{n+1}\setminus \{pt\}$.(There is a natural embedding$([x],v)\mapsto [x,x.v]$):

https://math.stackexchange.com/questions/1486105/is-the-total-space-of-the-tautological-line-bundle-over-mathbbrpn-a-non

Now assume that $E$ is the total space of the tautological 2-plane bundle over the real Grassmanian $G(2,n)$.

As a generalization of the above fact we ask:

Is there a (natural) embedding of $E$ into $G(2,n+1)$? If yes, what is the topological type of the remainder $G(2,n+1)\setminus E$. Is it possible to have an embedding with a one point remainder?

Answer 1

To answer the final question, assume that the total space of the taulogical bundle $E\to G(k,n)$ embeds into $G(k,n+1)$ such that $G(k,n+1)\setminus E$ consists of a single point only. Then $G(k,n+1)$ is the Thom space of $E$. The Thom isomorphism for $E$ shows that $H^\ell(G(k,n+1);\mathbb Z/2)=0$ for $0<\ell<k$.

On the other hand, the Grassmannian $G(k,n+1)$ approximates the classifying space $BGL(k,\mathbb R)$, and the map $G(k,n+1)\to BGL(k,\mathbb R)$ is $(n+1)$-connected. Hence, there is always a nontrivial homomorphism $\pi_1(G(k,n+1))\to\pi_1(BGL(k,\mathbb R))\cong\mathbb Z/2$, so $H^1(G(k,n+1);\mathbb Z/2)\ne 0$. But this implies that for $k\ge 2$, no embedding as above exists.

Answer 2

There's two kinds of $k$-planes in $\mathbb{R}^n\times \mathbb{R}$: Those that project isomorphically to $\mathbb{R}^n$, and those that contain $\mathbb{R}$.

The former are given as graphs of linear functions $V\rightarrow \mathbb{R}$, where $V\subset \mathbb{R}^n$ is a $k$-plane. With the canonical scalar product on $\mathbb{R}^n\times\mathbb{R}$ restricted to $V$, linear functions on $V$ naturally correspond to vectors. This means that we just constructed a bijection between tuples $(V,v), V\subset \mathbb{R}^n, v\in V$ and an open subset of $\operatorname{Gr}(k, n+1)$. The former is the usual description of the tautological bundle on $\operatorname{Gr}(k, n)$.

The complement (the planes that intersect $\mathbb{R}$) is in natural bijection to $\operatorname{Gr}(k-1,n)$, so we can see that that is your complement. In particular, you'll only get a point for $k=1$, the $\mathbb{R}P^n$-case.

Answer 3

$\newcommand{\bRP}{\mathbb{RP}}$ $\newcommand{\bR}{\mathbb{R}}$ The result about the complement of a point in $\bRP^{n+1}$ is related to the natural cell decomposition of $\bRP^{n+1}$. The counterpart of this decomposition for higher Grassmannians is the so called Schubert decomposition and you can find a particularly readable description in Chapter 6 of Milnor & Stasheff's classic Characteristic Classes.

These cell decompositions have a Morse theoretic description, and this point of view will enable you to construct embeddings of many homogeneous spaces in to Grassmannians.

Fix an $n$-dimensional dimensional Euclidean space $V$ and denote by $\DeclareMathOperator{\Gr}{\boldsymbol{Gr}}$ $\Gr_k(V)$ the Grassmanian of $k$-dimensional subspaces of $V$. $\DeclareMathOperator{\Sym}{Sym}$. For a subspace $S\in\Gr_k(V)$, denote by $P_S$ the orthogonal projection onto $S$ viewed as a symmetric operator $P_S: V\to V$. Denote by $\Sym(V)$ the space of symmetric linear operators $V\to V$.

The correspondence

$$\Gr_k(V)\ni S\mapsto P_S\in\Sym(V) $$

produces a smooth embedding $\Gr_k(V\hookrightarrow \Sym(V)$.

The space $\Sym(V)$ is equipped with a natural inner product $\DeclareMathOperator{\tr}{tr}$

$$(A,B)=\tr(AB),\;\;\forall A,B\in \Sym(V). $$

This induces a Riemann metric on $\Gr_k(V)$.

Any operator $ A\in \Sym(V)$ defines a linear function $\ell_A:\Sym(V)\to\bR$, $B\mapsto \tr(AB)=(A,B)$. We denote by $f_A$ the restriction of $\ell_A$ to $\Gr_k(V)$.

For generic $A$ the function $f_A:\Gr_k(V)\to \bR$ is a Morse function. We denote by $\nabla f_A$ the gradient of $f_A$ with respect to the induced metric and by $\Phi_A^t$ the flow on $\Gr_k(V)$ generated by $-\nabla f_A$. Assuming $A$ generic, i.e., it has distinct eigenvalues, then the unstable manifolds of this flow are precisely the Schubert cells giving the Schubert cellular decomposition described by Milnor and Stasheff.

When $A$ is not generic $f_A$ is not necessarily Morse but it is Morse-Bott. In this case the critical submanifolds of $f_A$ are intersting homogeneous spaces. For example, if you take $A$ to be the orthogonal projection on a $1$-dimensional subspace $L$,then the restriction of $\ell_A$ to $\Gr_1(V)$ is Morse-Bott. Its absolute minima form a critical submanifold diffeomorphic to $\Gr_1(L^\perp)$, where $L^\perp$ is the orthogonal complement of $L$ in $V$. This function has a unique maximum, the point $L\in\Gr_1(V)$. From these two facts you get the statement about the complement of a point in $\Gr_1(V)$ mentioned at the begining of your question.

One can use the same function

$$\Gr_2(V)\ni S\mapsto \tr(P_LP_S)\in\bR $$

to obtain other interesting embeddings. For more details and other examples see this very nice article by Dynnikov and Veselov and Chapter 3 of my book on Morse theory.

Update 1. Here is an answer to your question. The Grassmannian $\Gr_2(\bR^{n-1})$ embeds in $\Gr_2(\bR^n)$. Using the above notation observe that $\Gr_2(L^\perp)$ ($2$-planes in $L^\perp$) embeds in $\Gr_2(V)$. The normal bundle of this embedding is the tautological $2$-plane bundle over $\Gr_2(L^\perp)$. This submanifold consists of the minima of the function $f_A$ where $A=P_L$. The complement of tubular neighborhood is not a disk though.

The maxima of the function $f_A$ consists of $2$-planes containing $L$. It is not hard to see that this set can be identified with lines in $V$ perpendicular to $L$, i.e., $\Gr_1(L^\perp)$. The normal bundle of this embedding is quotient tautological bundle, i.e.,the quotient of the trivial bundle $$ L^\perp\times \Gr_1(L^\perp)\to\Gr_1(L^\perp)$$ by the universal line bundle over $\Gr_1(L^\perp)$. Since the critical points of the function $f_A$ are either global minima or global maxima we deduce shows that $\newcommand{\bD}{\mathbb{D}}$

$$\Gr_2(\bR^n)= \bD_{\Gr_2(\bR^{n-1})}\cup_\partial \bD_{\Gr_{n-2}(\bR^{n-1})}, $$

where $\bD_{\Gr_k(V)}$ denotes the unit disk bundle of the tautological vector bundle over $\Gr_k(V)$, and $\cup_\partial$ denotes the gluing of two manifolds along their diffeomorphic boundaries.