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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!

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What do you mean with the brackets $\langle...\rangle$? May be the span? –  Daniele Zuddas May 5 '12 at 13:45
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Some remarks and questions: 1. You can make $\mathrm{im}W$ canonical by replacing $L^c$ with he quotient $V/L$. Then $T_{L}G\cong\mathrm{Hom}(L,V/L)$ with no choices and $\mathrm{im}W\leq V/L$ makes sense and has a well-defined dimension. This answers part of question A. 2. Do you mean $W\leq T_L G$ when you say $W\leq T_G$? –  Fran Burstall May 6 '12 at 10:40
    
I do not understand the definition of $\mathrm{osc}W$: apparently, this should be a subspace of $\mathrm{Hom}(L,L^c)$ while $\langle L, \mathrm{im} W\rangle$ is a subspace of $V$. –  Fran Burstall May 6 '12 at 10:49
    
@Daniele: yes, I mean the span. @Fran: sorry for 2 misprints (just fixed them)! I'm aware that I can meke $\mathrm{im}W$ canonical by using $V/L$, but, for my purposes, $\mathrm{im}W$ is just an intermediate step to define $\mathrm{osc}W$, which is a subspace of $V$, whose dimension exceed that of $L$ precicely by $\mathrm{rank}W$. I'm interested in these osculators since they seem to provide a link between Grassmannians of different order: if I'm not mistaken, the family of tangent lines of rank 1 on $Gr(V,n)$ is fibered over $Gr(V,n+1)$. –  G_infinity May 8 '12 at 19:13
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If I view $\mathrm{im}W$ as a subspace of $V/L$, then $\mathrm{osc}W$ is simply the preimage of $\mathrm{im} W$ by the quotient map $V\to V/L$. I guess that makes $\mathrm{osc}W$ transparently canonical. –  Fran Burstall May 10 '12 at 19:25

1 Answer 1

up vote 3 down vote accepted

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.

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@Robert: Thank for the time you've devoted to write down this. Now it's more clear to me what I had in mind! Just a comment: we can speak of "osculation map" $o$, even without a submanifold $M$ of Gr$(V,n)$. What we need is a "contant-rank" distribution $D$ on Gr$(V,n)$, i.e., such that osc$D_L$ has constant dimension $n+r$, for all $L\in$Gr$(V,n)$. Then $o_D:L\mapsto$osc$D_L$ is a map from Gr$(V,n)$ to Gr$(V,n+r)$. My idea was to somehow "invert" such an oculatin map, starting from a submanifold $\Delta\subseteq$Gr$(V,n+r)$, and look for a distribution $D$ s.t. the image of $o_D$ is $\Delta$ –  G_infinity Jul 12 '12 at 6:37
    
... of course, as you noticed, such an assigment is not unique. In fact, I had in mind a way to make it unique. I needed a map σ:Gr$(V,n+r)\mapsto$Gr$(V,n)$, s.t. $K\supseteq σ(K)$, for all $K\in$Gr$(V,n+r)$, i.e., a splitting of the projection of Fl$(V,n,n+r)$ over Gr$(V,n+r)$, and also that $\sigma(\Delta)$ is integral w.r.t the canonical distribution. With this additional datum, I can associate with any $L\in$Gr$(V,n)$ the (now) unique subspace $W\leq T_L$Gr$(V,n)$, s.t. osc$W=K\in\Delta$ AND $\sigma(K)=L$, obtaining a distribution D, s.t. (if I'm not mistaken) im$o_D=\Delta$. –  G_infinity Jul 12 '12 at 6:59
    
Dear @RobertBryant, do you happen to know if the distribution on partial flags you describe here appears in the literature or has a name? I would like to make a reference to it. (Of course I could also use your answer as reference). –  Michael Bächtold Jul 17 '13 at 18:16

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