# Tautological and normal bundles over flag manifolds and jet bundles

Hello! Recently, doing my research on jet bundles, I was led to consider the following construction.

Let $V$ be a real vector space of dimension $n$. Consider the flag manifold $G(V,k,l)$ and the two Grassmann manifolds $G(V,k)$ and $G(V,l)$. It seems to me that there are two natural projections $\eta:G(V,k,l)\to G(V,k)$ and $\pi:G(V,k,l)\to G(V,l)$. If I'm not mistaken, $\eta^{-1}(K)=G(K,l)$ and $\pi^{-1}(L)=G((V/L)^*,k-l)$. Now each Grassmann manifold is accompanied by its own tautological bundle, and I can use the above projections to lift the tautological bundles over $G(V,k,l)$. Namely, let $\Theta$ (resp., $\theta$) the tautological bundle over $G(V,k)$ (resp., over $G(V,l)$), and consider their pullbacks $\overline{\Theta}:=\eta^\ast(\Theta)$ and $\overline{\theta}:=\pi^\ast(\theta)$. By definition, the $\overline{\Theta}$-fiber over a point $(K,L)\in G(V,k,l)$ is $K$, while the $\overline{\theta}$-fiber over the same point is $L$. Hence, I can obtain a "normal tautological bundle" $\nu:=\overline{\Theta}/\overline{\theta}$ of rank $k-l$.

QUESTION A

• Is the bundle $\nu$ I've just described a well-known construction? If yes, where I can find it?
• In the case of a complete flag manifolds, we would have an $n$--tuple of line bundles. Does it have any special interpretation?

Now let $\pi:E\to M$ a smooth vector bundle of smooth manifolds, of rank $n$. It seems natural to associate to $\pi$ a smooth bundle $G(\pi,k,l)$ by replacing each fiber with its flag (or, in particular, Grassmann) manifold. In its turn, $G(\pi,k,l)$ is the base of vector bundles of tautological type (constructed, fiber by fiber, just as above). I'm aware of a case when this construction is of some use: $G(\tau_M,k)$ is the 1-st jet bundle $J^1(M,k)$ over the $k$--dimensional manifold $M$ ($\tau_M$ being the tangent bundle); the corresponding tautological $k$-dimensional bundle over $J^1(M,k)$ is by some authors called the $R$-distribution, and it plays a key role in defining higher order jet spaces and the Cartan distribution over them.

QUESTION B

• Is the construction of "flag bundles" out of vector ones a well-known one? If yes, are their properties (e.g., behavior w.r.t. bundle morphisms) examined somewhere?
• Does exist an analogous construction of jet spaces, carried out by using the "flag bundles" of the tangent bundle, instead of the Grassmann bundles? If yes, does the "normal tautological bundle" (describe above) play any relevant role in such a theory?