Uniform bundles on quadrics

We show that there exist only constant morphisms from $\mathbb{Q}^{2n+1}(n\geq 1)$ to $\mathbb{G}(l,2n+1)$ if $l$ is even $(0<l<2n)$ and $(l,2n+1)$ is not $(2,5)$. As an application, we prove on $\mathbb{Q}^{2m+1}$ and $\mathbb{Q}^{2m+2}(m\geq 3)$, any uniform bundle of rank at most $2m$ splits, which improves the upper bound of splitting for uniform bundles obtained by Kachi and Sato. We classify all unsplit uniform bundles of minimal rank on $B_n/P_k$ $(k=\frac{2n}{3},k\ge6)$ and $D_n/P_k$ $(k=\frac{2n-2}{3},k\ge 6)$. We partially answer a conjecture of Ellia, which predicts that some uniform bundles of special splitting types on $\mathbb{P}^n$ necessarily split and we find some restrictions on the splitting types of unsplit uniform bundles of minimal rank.