Let $X$ be a rational homogeneous space. It is well known that $X$ can be embedded in a projective space so that it is covered by lines. A vector bundle on $X$ is said to be uniform if its restriction to any line is the same.
Given a vector bundle $E$ on $X$, a point $x\in X$, and a line $\ell\subset X$ through $x$, one can construct in a natural way a flag on the fiber of $E$ at $x$: $$ E^1_{x,\ell}\subset E^2_{x,\ell} \subset \cdots \subset E^k_{x,\ell}=E_x. $$ When the vector bundle $E$ is uniform, the dimensions $d_i=\dim E^i_{x,\ell}$ do not depend on the choice of the line $\ell$. So one gets a morphism: $$ s_{E,x}:H_x\to F(d_1, d_2, \dots, d_k; E_x) $$ from the space $H_x$ of lines on $X$ through $x$ to the appropriate flag variety.
This morphism encodes geometric properties of $E$. For instance, we show that the morphism $s_{E,x}$ is constant if and only if $E$ splits as a sum of line bundles. This result generalizes and provides a unified proof of several splitting criteria for uniform vector bundles on rational homogeneous spaces.
This is a joint work with Nicolas Puignau.