Erratum to: On the geometry of spaces of oriented geodesics

Abstract

1. Those entries of Table 1 in the original paper listing the numbers of geometric structures on the space of geodesics L (S p,q) of the 3-dimensional space S p,q of non-zero constant curvature with p+q = 3 are incorrect. In this note we give the correct values and arguments. In original paper we failed to note that the (so(V )+so(V ′))-module V ⊗V ′, where V, V ′ are 2-dimensional pseudo-Euclidian spaces, has not only the invariant (para)complex structures J = JV ⊗ 1, J ′ = 1 ⊗ JV ′ , but that also their product J ′′ = J J ′ = JV ⊗ JV ′ is an invariant (para)complex structure. We next analyze the impact of this observation in detail and correct Table 1 accordingly. The authors would like to thank Henri Anciaux for drawing their attention to this oversight. 2. Let E = E p+1,q be a pseudo-Euclidean 4-dimensional vector space of signature (p + 1, q) = (4, 0), (3, 1), (2, 2) or (1, 3) with an orthonormal basis e1, e2, e3, e4 with e2 4 := g(e4, e4) = 1. We denote by E ′ = e⊥ 4 the orthogonal complement to e4. The pseudo-sphere