Projected vector transport on Stiefel and Grassmann

Let $X$ be a source frame, $\eta \in T_X\mathrm{St}(n,r)$, and $Z\in T_X\mathrm{St}(n,r)$. For Stiefel transport let $Y=R_X(\eta )$ be either the QR or polar endpoint. For Grassmann transport, require $\eta$ to be the unique horizontal representative at $[X]$, so $X^{\mathsf{T}}\eta =0$, and choose $Y=R_X^{\mathrm{p}\mathrm{o}\mathrm{l}}(\eta )$, the polar endpoint; the positive-diagonal QR frame is not used for this representative-based quotient transport. Define the projected Stiefel transport ${\mathcal{T}}_Y^{\mathrm{S}\mathrm{t}}(Z):=Z-Y\mathrm{sym}(Y^{\mathsf{T}}Z),$ and, for a horizontal source representative $Z$, define the projected Grassmann transport representative ${\mathcal{T}}_Y^{\mathrm{G}\mathrm{r}}(Z):=(I-Y{Y}^{\mathsf{T}})Z\mathrm{.}$ The Grassmann expression is interpreted as a tangent representative at $[Y]$. The polar endpoint is right-equivariant, so the next proposition verifies representative independence, tangency, and identity at a zero step. These projection transports are not claimed to be Levi-Civita parallel transport.