Real Stiefel and Grassmann optimization conventions

For any real matrix $U$, $⟨U,U{⟩}_F={\sum }_{i,j}{U}_{ij}^2$, so this pairing is symmetric bilinear and positive for every nonzero $U$ by [F3]. Its restriction to each tangent or horizontal subspace is therefore a positive-definite inner product, as required by [F2].

In a local Grassmann graph chart, choose a smooth orthonormal frame section $X(u)$ from the local frame-bundle trivialization in [F1], and set $P(u)=X(u)X(u{)}^{\mathsf{T}}$. For a chart tangent $v$, differentiating $X(u{)}^{\mathsf{T}}X(u)=I$ gives $A=X(u{)}^{\mathsf{T}}D{X}_u[v]$ skew-symmetric. The first term $X(u)A$ is vertical: for every skew-symmetric $A$, the Cayley curve $Q(t)=(I_r-\frac{t}{2}A{)}^{-1}(I_r+\frac{t}{2}A)$ satisfies $Q(t{)}^{\mathsf{T}}Q(t)=I_r$ near zero and $Q(t)=I_r+tA+O(t^2)$. Thus $Z=(I-X(u)X(u{)}^{\mathsf{T}})D{X}_u[v]$ is the unique horizontal representative of $v$. Then $D{P}_u[v]=ZX(u{)}^{\mathsf{T}}+X(u)Z^{\mathsf{T}}$. For horizontal $Z,W$, direct expansion gives $\frac{1}{2}⟨Z{X}^{\mathsf{T}}+X{Z}^{\mathsf{T}},W{X}^{\mathsf{T}}+X{W}^{\mathsf{T}}{⟩}_F=⟨Z,W{⟩}_F$. Thus the quotient pairing has smooth coefficients in every graph chart. In a local product chart of the frame bundle, matrix entries vary smoothly, so the ambient Frobenius pairing has smooth coefficients on Stiefel as well.

By [F4], each frame represents an $r$-dimensional subspace, and [F1] identifies frames representing the same subspace by the right orthogonal action. For $Q^{\mathsf{T}}Q=I_r$, cyclicity of trace gives $⟨UQ,VQ{⟩}_F=\mathrm{tr}(Q^{\mathsf{T}}{U}^{\mathsf{T}}VQ)=\mathrm{tr}(U^{\mathsf{T}}V)$. Thus the action is isometric and the horizontal pairing is independent of the chosen orthonormal frame.