How can I equip the universal vector bundle over $\mathbf{G}(k,n)$ with a connection?

Question

When constructing the grassmannians $\mathbf{G}_\mathbb{F}(k,n)$ for $\mathbb{F} = \{\mathbb{R},\mathbb{C} \}$ there is a natural vector bundle $$ \mathbf{G}_{k,n} \to \mathbf{G}_\mathbb{F}(k,n) $$ whose fiber over a point $[V]$ is the vector space $V$. Is there a natural connection I can equip to this bundle? Also, in the complex case, is there a natural holomorphic connection?

Answer 1

Yes, do the analogue of the Levi-Civita connection for submanifolds of Euclidean space. Differentiate a section (as an $\Bbb F^n$-valued function) and project orthogonally onto the subspace.