Question

Hello! Recenlty, doing my research, I came across a quite natural construction, and I would like to know more about it. Unfortunately, being not expert neither in Grassmannians nor in Contact Geometry, I wasn't able to find anything in literature - but I'm confindent that an expert in these area will easily help me.

Suppose $G=Gr(V,n)$ is the (real) Grassmannian of a vector space $V$. Suppose also that we are able to find a complement $L^c$ to any element $L\in G$ (e.g., by equipping $V$ with a metric). Then $\mathrm{Hom}(L,L^c)$ is an open neighborhood of $L$, canonically identified with $T_L G$. Given a linear subspace $W\leq T_LG$, it is natural, for me, to define the following subspaces:

The "kernel" of $W$, defined as $\ker W:=\cap_{h\in W}\ker h\leq L$.

The "image" of $W$, defined as $\mathrm{im}W:=\langle h(L)\mid h\in W\rangle\leq L^c$.

The "osculator" of $W$, defined as $\mathrm{osc}W:=\langle L, \mathrm{im}W\rangle\leq V$.

If I'm not mistaken, $\mathrm{osc}W$ admits the following geometrical interpretation: $W$ determines, up to first order of tangency, a $\dim W$-parametric family of $n$-dimensional subspaces of $V$, whose enveloping surface has $\mathrm{osc}W$ as its tangent space at $L$. As such, $\mathrm{osc}W$ is canonical (by "canonical" I mean here that it doesn't depend on the choice of $L^c$).

QUESTION A: is $\ker W$ canonical too? if yes, what about an its geometrical interpretation? of course $\mathrm{im}W$ is not canonical, but is its dimension (denoted by $\mathrm{rank}W$) canonical?

Incidentally, if anyone can point me to some book/paper where this stuff is described, I'd be grateful.

I was interested in osculators, since I noticed that a submanifold $\Delta\subseteq Gr(V,n+r)$ determines a distribution of rank-$r$ tangent subspaces on $G$. Indeed, for any point $L\in G$, I can declare that a subspace $W\leq T_LG$ belongs to the distribution, iff $\mathrm{rank}W=r$ and $\mathrm{osc}W\in\Delta$.

In particular, I was palying with $\mathbb{P}V=Gr(V,1)$, where $\dim V=4$, and a submanifold $\Delta\subseteq G=Gr(V,2)$, which determided, in the sense explained above, a rank-1 distribution, which turned out to be a contact one on $\mathbb{RP}^3$.

QUESTION B: is this construction of a distribution a well-known fact? if yes, which conditions has $\Delta$ to satisfy, in order to have a smooth distribution? in the more specific case of $\dim V=4$, can I recognize that a distribution on $\mathbb{P}V$ is a contact one, just by looking at the corresponding $\Delta$ in $G$?

I'm really unable to find references, so any help will be welcome!

Answer 1

I think you want to have a look at the following paper:

P.A. Griffiths & J. Harris, Algebraic Geometry and Local Differential Geometry, Ann. Scient. Ec. Norm. Sup. 12 (1979) 355--432, MR0559347.

This will answer a lot of your questions about the geometry of submanifolds of Grassmannians and, in particular, the linear algebra of subspaces of their tangent spaces. In particular, the subspaces $\ker (W)\subset L$ and $\text{im}(W)\subset V/L$ for $W\subset T_LG$ make appearances, though I'm not sure the names are the same. (It has been a while since I looked at that paper.)

One thing to note is that, if $M\subset \text{Gr}(V,n)$ is a submanifold such that $\ker(T_LM)\subset L$ has constant dimension, say $\nu_M < n$, for $L\in M$, then the assignment $L\mapsto \ker(T_LM)$ defines a smooth map $\kappa:M\to\text{Gr}(V,\nu_M)$. This map need not be an immersion; it could even be constant, as is often the case when $M$ lies in the submanifold of $\text{Gr}(V,n)$ consisting of those $n$-planes that contain a fixed $\nu_M$-plane $S\in \text{Gr}(V,\nu_M)$. (In this special case, one can regard $M$ as a submanifold of $\text{Gr}(V/S,n{-}\nu_M)$, and the geometry could be easier to understand there.)

Another thing you can observe is that if $M\subset \text{Gr}(V,n)$ is a submanifold such that $\text{osc}(T_LM)\subset V$ has constant dimension, say $\mu_M > n$, for $L\in M$, then the assignment $L\mapsto \text{osc}(T_LM)$ defines a smooth map $\omicron: M\to\text{Gr}(V,\mu_M)$. The image need not be a smooth manifold, but, when it is, it has to satisfy $\nu_{\omicron(M)} \ge n$. If the image is constant, say $\text{osc}(T_LM) = K$ for all $L\in M$, then, of course, one has $M\subset \text{Gr}(K,n)$.

I don't understand your 'definition' of a 'distribution' on $\text{Gr}(V,n)$ associated to a submanifold $\Delta\subset\text{Gr}(V,n{+}r)$. Surely it's not common for an $r$-dimensional subspace $W\subset \text{Hom}(L,V/L)$ to have $n{+}r$ as the dimension of $\text{osc}(W)$, is it? This sounds very special to me, and I don't think it's likely, for most $\Delta$ that there would be only one such subspace $W\subset \text{Hom}(L,V/L)$ for each $L\in \text{Gr}(V,n)$.

It is likely that you are dealing with a more general differential system than a 'distribution'. In fact, I think you should probably be thinking in terms of the partial flag variety $\text{Fl}(V;n,n{+}r)\subset\text{Gr}(V,n)\times\text{Gr}(V,n{+}r)$ consisting of those pairs $(L,K)$ where $L\in \text{Gr}(V,n)$ and $K\in \text{Gr}(V,n{+}r)$ satisfy $L\subset K$. This is a smooth manifold that fits into a double fibration picture $$ \begin{matrix} & & \text{Fl}(V;n,n{+}r) & & \\ &\swarrow & & \searrow & \\ \text{Gr}(V,n) & & & & \text{Gr}(V,n{+}r) \end{matrix} $$ and its tangent space at each point $(L,K)$ consists of those $(a,b)\in \hom(L,V/L)\times\hom(K,V/K)$ such that $a(\ell)\equiv b(\ell)\mod K$ for all $\ell\in L$. , There is also a canonical subspace $D_{(K,L)}\subset T_{(K,L)}\text{Fl}(V;n,n{+}r)$ consisting of those $(a,b)\in \hom(L,V/L)\times\hom(K,V/K)$ such that $b(\ell) = a(\ell)\mod K = 0$ for all $\ell\in L$. This defines a distribution on $\text{Fl}(V;n,n{+}r)$ such that the lifting of $M\subset\text{Gr}(V,n)$ into $\text{Fl}(V;n,n{+}r)$ defined by the assignment $L\mapsto \bigl(L,\text{osc}(T_LM)\bigr)$ is tangent to $D$ everywhere.

Now, if one fixes a submanifold $\Delta\subset \text{Gr}(V,n{+}r)$ and considers the submanifold $\hat\Delta\subset \text{Fl}(V;n,n{+}r)$ consisting of those pairs $(L,K)$ with $K\in\Delta$, then taking the subset of tangent vectors in $D$ that are tangent to $\hat\Delta$ defines a distribution on $\hat\Delta$ such that the manifolds tangent to this distribution are the objects you want to study. They contain the lifts (in the above sense) of the manifolds $M\subset \text{Gr}(V,n)$ whose osculation maps land you in $\Delta$.

At this point, you'll need the theory of exterior differential systems to understand the 'generality' of those submanifolds tangent to $D$ that lie in $\hat\Delta$. For small values of $n$ and $r$, this will be easy to understand by the methods of contact and symplectic geometry, but as soon as they get larger and $\Delta$ has high codimension, you'll need more powerful methods.