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3 edited tags
2 I fixed two misprints: $T_LG$ instead of $T_G$ and $\mathrm{osc}W:=\langle L, \mathrm{im}W\rangle\leq V$ instead of ...$\leq W$

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 $W\leq T_G$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 W$V$.

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!

1

# Osculating spaces and distributions on (real) Grassmannian manifold

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 $W\leq T_G$, 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 $\mathrm{osc}W:=\langle L, \mathrm{im}W\rangle\leq W$.

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!