Embeddings of Grassmann graphs

Let $V$ and $V^{\mathrm{\prime }}$ be vector spaces of dimension $n$ and $n^{\mathrm{\prime }}$, respectively. Let $k\in \{2,\mathrm{.}\mathrm{.}\mathrm{.},n-2\}$ and $k^{\mathrm{\prime }}\in \{2,\mathrm{.}\mathrm{.}\mathrm{.},n^{\mathrm{\prime }}-2\}$. We describe all isometric and $l$-rigid isometric embeddings of the Grassmann graph ${\mathrm{\Gamma }}_k(V)$ in the Grassmann graph ${\mathrm{\Gamma }}_{k^{\mathrm{\prime }}}(V^{\mathrm{\prime }})$.