Let $X$ be a smooth $n+m$ dimensional manifold and $\pi:Y\rightarrow X$ the bundle whose fibre over any $x\in X$ is the Grassmann manifold of all $m$ dimensional subspaces of $T_xX$. This is equivalent to the manifold $J^1(X,m)$ of all first order contact elements of dimension $m$, also called the space of order $1$ jets of $m$ dimensional submanifolds in $X$.
I am interested in knowing the de Rham cohomology $H^\ast_{\mathrm{dR}}(Y)$ of the total space of this bundle in terms of the de Rham cohomology $H^\ast_{\mathrm{dR}}(X)$ of $X$.
I assume this is computable and have been computed, but I cannot find it anywhere and I am not familiar enough with algebraic topology to compute it myself.