Let $\mathbb K$ be either $\mathbb R$ or $\mathbb C$, let $n\ge 1$, and let $0\le k\le n$. Let $\operatorname{Gr}(k,n)$ be the smooth Grassmannian of $k$-dimensional $\mathbb K$-linear subspaces of $\mathbb K^n$ in the [orthogonal projection](/theorems/437) model, and let

\begin{align*} P:\operatorname{Gr}(k,n)\to \operatorname{End}_{\mathbb K}(\mathbb K^n) \end{align*}

be the smooth map sending a subspace to its orthogonal projection. Let

\begin{align*} S:=\{(P_0,v)\in \operatorname{Gr}(k,n)\times \mathbb K^n:P_0v=v\} \end{align*}

be the tautological rank-$k$ vector bundle over $\operatorname{Gr}(k,n)$. Equip $S$ with the universal connection

\begin{align*} \nabla s:=P\,ds \end{align*}

for every local smooth section $s$ of $S$, where $s$ is viewed as a $\mathbb K^n$-valued function through the inclusion $S\subset \operatorname{Gr}(k,n)\times\mathbb K^n$. Extend $\nabla$ to $S$-valued differential forms by representing them as $\mathbb K^n$-valued forms in the trivial bundle and applying $\alpha\mapsto P\,d\alpha$. Then the curvature $F_\nabla$, defined by $F_\nabla s=(d^\nabla\nabla s)$ for local sections $s$, is the $\operatorname{End}(S)$-valued $2$-form

\begin{align*} F_\nabla=P\,dP\wedge dP\,P, \end{align*}

where multiplication denotes composition of endomorphism-valued forms together with exterior product of form entries, restricted to the tautological subbundle $S$.