Topology of Real Schlafli Six-Line Configurations on Cubic Surfaces and in mathbb{RP}³

A famous configuration of 27 lines on a non-singular cubic surface in $\mathbb P^3$ contains remarkable subconfigurations, and in particular the ones formed by six pairwise disjoint lines. We study such six-line configurations in the case of real cubic surfaces from topological viewpoint, as configurations of six disjoint lines in the real projective 3-space, and show that the condition that they lie on a cubic surface implies a very special property of {\it homogeneity}. This property distinguish them in the list of 11 deformation types of configurations formed by six disjoint lines in $\mathbb{RP}^3$.