The Grassmann manifold $\widetilde{Gr}(k,\Bbb{R}^n)$ of oriented $k$-planes in $\Bbb{R}^n$ is a double cover of the Grassmann manifold $Gr(k,\Bbb{R}^n)$ of non-oriented $k$-planes. We can give $\widetilde{Gr}(k,\Bbb{R}^n)$ the covering metric making the covering a local isometry.

Is it possible to describe the geodesics in $\widetilde{Gr}(k,\Bbb{R}^n)$ simply in terms of the geodesics in the non-oriented $Gr(k,\Bbb{R}^n)$?