Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 115 / 3

A principal connection on $P$ is a $G$-equivariant smooth splittingwhere $V_p$ is tangent to the $G$-orbit. Equivalently, it is a $\mathfrak{g}$-valued one-form $\mathcal{A}$ satisfying $\mathcal{A}({\xi }_P)=\xi$ and $R_g^{\ast }\mathcal{A}={\mathrm{Ad}}_{g^{-1}}\mathcal{A}$. Its curvature of a principal connection is

On $S^n\times S^n$, consider $q(x,y)=⟨x,y⟩$. At a point of $q^{-1}(0)$, its differential isThe tangent vector $(y,x)$ lies in $T_x{S}^n\oplus T_y{S}^n$ and has $dq(y,x)=2$, so zero is a regular value. The regular level set theorem makes $P=q^{-1}(0)$ a smooth submanifold. Its tangent space is

The points $(x,y)$ are ordered orthonormal two-frames, so $P$ is the Stiefel manifold $V_2({\mathbb{R}}^{n+1})$. Changing an orthonormal basis of the same plane gives the displayed free right $O(2)$-action. Since $O(2)$ is compact, the action is proper, and the quotient is the Grassmannian of unoriented two-planes. The Free proper Lie-group action theorem makesa principal $O(2)$-bundle.

LetThe vertical space is spanned by the fundamental vector $(y,-x)$ corresponding to $J\in \mathfrak{o}(2)$. DefineIt sends $(y,-x)$ to $J$, is $O(2)$-equivariant, and therefore is a principal connection. Its kernel consists exactly of those $(u,v)$ for whichThese are precisely the velocities satisfying the stated horizontality condition. Every tangent vector has the unique decompositioninto vertical and horizontal parts, proving uniqueness. This is the Canonical principal connection on the Stiefel bundle over a Grassmannian.

Write $\mathcal{A}=aJ$ with the scalar one-formThe Lie algebra $\mathfrak{o}(2)$ is abelian, so $[\mathcal{A}\wedge \mathcal{A}]=0$. MoreoverConsequentlyas required.