Totally geodesic subsets in the variety of directions of physical space

Let $M_0$ be a Minkowski 4-spase, $\Lambda_2(M_0)$ its second exterior power equipped with a structure of pseudo-Euclidean space with singature $(3,3)$, $K_0(M_0)$ the light cone, $G_1\subset\Lambda_2(M_0)$ the set of oriented 2-planes meeting the interior of $K_0(M_0)$. In the paper, 4 types of totally geodesic two-manifolds in $G_1$ are discribed, such that manifolds of one type are pairwise congruent as subsets in $\Lambda_2(M_0)$, while mainfolds of different types are not. Models of such mainfolds in the disk $D^3$ are constructed. An explicit formula for the curvature of $G_1$ is given.