Erratum to: On the geometry of spaces of oriented geodesics

1 Erratum to: Ann Glob Anal Geom (2011) 40:389–409 DOI 10.1007/s10455-011-9261-5

We correct some of the entries of Table 1 in our original paper.

1. Those entries of Table 1 in the original paper listing the numbers of geometric structures on the space of geodesics $L^{\epsilon }(S^{p,q})$ 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 $(\mathfrak {so}(V) + \mathfrak {so}(V'))$-module $V \otimes V'$, where $V,V'$ are 2-dimensional pseudo-Euclidian spaces, has not only the invariant (para)complex structures $J = J_V \otimes 1,\, J' = 1 \otimes J_{V'}$, but that also their product $J'' = J J' = J_V \otimes J_{V'}$ 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 $e_1,e_2,e_3, e_4$ with $e_4^2 := g(e_4,e_4)=1$. We denote by $E' = e_4^{\perp }$ the orthogonal complement to $e_4$. The pseudo-sphere of unit vectors $S^{p,q} = \{x \in E,\, x^2=1\}$ is the orbit $S^{p,q} = SO(E) e_4$. It has the reductive decomposition

$$\begin{aligned} \mathfrak {so}(E) = \mathfrak {so}(E') + e_4\wedge E' , \end{aligned}$$

where $\mathfrak {so}(E') \simeq \mathfrak {so}_{3,0}, \mathfrak {so}_{2,1}, \mathfrak {so}_{1,2}, \mathfrak {so}_{0,3}$, respectively. The metric of E induces an $SO(E')$-invariant metric in the tangent space $T_0S^{p,q} = e_4 \wedge E'$. The stability subalgebra of the geodesic $\ell (e_4 \wedge e_3)$ through the origin $o =e_4 \in S^{p,q}$ in the direction $e_4 \wedge e_3 \in T_o S^{p,q} = e_4 \wedge E'$ is $\mathfrak {so}(V) \times \mathfrak {so}(V^{\perp })$, where $V = \mathrm {span}(e_1, e_2)$ and $V^{\perp } = \mathrm {span}(e_3,e_4)$. The geodesic is spacelike if $\epsilon = e_3^2 =1$ and timelike if $\epsilon = e_3^2 =-1$. The space of such geodesics is identified with $L^{\epsilon }(S^{p,q}) = SO(E)/ SO(V) \times SO(V^{\perp })$ (up to a covering).

We now denote by $g_V$ the unique (up to a scale) metric in 2-dimensional SO(V)-module V and by $\omega _V$ unique ( up to scale ) 2-form. Then for some positive $\lambda$, the complex structure $J_V = \lambda g_V^{-1} \circ \omega _V$ is the unique (up to sign) $\mathfrak {so}(V)$-invariant complex structure if the metric $g_V$ has signature (2, 0) or (0, 2). It is the unique (up to sign) paracomplex structure if the signature of $g_V$ is (1, 1). We use similar notations for $V^{\perp }$. The reductive decomposition of the symmetric space $L^{\epsilon }(S^{p,q})$ is given by

$$\begin{aligned} \mathfrak {so}(E) = (\mathfrak {so}(V)\oplus (\mathfrak {so}(V^{\perp })) + V \wedge V^{\perp } . \end{aligned}$$

The tangent space $T_o L^{\epsilon }(S^{p,q})$ is identified with $\mathfrak {m}:= V \wedge V^{\perp } \simeq V \otimes V^{\perp }$ with the natural action of the isotropy algebra $\mathfrak {h}= \mathfrak {so}(V) + \mathfrak {so}(V^{\perp })$. We have therefore proved the following.

Table 1 Invariant geometric structures

Proposition

The $\mathfrak {h}$-invariant symmetric bilinear form (respectively, invariant 2-form) on $\mathfrak {m}$ are of the form

$$\begin{aligned} h_{\lambda , \mu }= \lambda \omega _V \otimes \omega _{V^{\perp }} + \mu g_V \otimes g_{V^{\perp }} ,\;\; \omega _{\lambda , \mu } = \lambda g_V \otimes \omega _{V^{\perp }} + \mu \omega _V \otimes g_{V^{\perp }} \end{aligned}$$

for $\lambda , \mu \in \mathbb {R}$.

The endomorphisms $J:= J_V \otimes 1, J^{\perp } := 1 \otimes J_{V^{\perp }}, J' := J \circ J^{\perp } = J_V \otimes J_{V^{\perp }}$ are invariant (para)complex structure in $\mathfrak {m}$. Moreover, $J.J^{\perp }$ are skew-symmetric with respect to any metric $h_{\lambda , \mu }$ and the endomorphism $J'$ is symmetric.

Since $S^{p,q}$ is an irreducible symmetric space, any invariant tensor field is parallel. We obtain

Corollary

Any non-degenerate form $h_{\lambda , \mu }$ defines an invariant pseudo-Riemannian metric, any non-degenerate 2-form $\omega _{\lambda , \mu }$ defines an invariant symplectic form and the endomorphisms $J,J^{\perp }, J'$ define integrable (para)complex structures on $L^{\epsilon }(S^{p,q})$. Furthermore, any pair of the form $(h_{\lambda , \mu }, J)$ or $(h_{\lambda .\mu }, J^{\perp })$, where $h_{\lambda , \mu }$ is as above , defines a (para)Kähler structure on $\mathfrak {m}$.

3. In the following table we indicate the signature of the metrics $g_V$ and $g_{V^{\perp }}$ for different values of p and $\epsilon$ and the type of the endomorphisms $J= J_V \otimes 1, \, J^{\perp } = 1 \otimes J_{V^{\perp }}, J'= J_V \otimes J_V{\perp }$ ( cx $=$ complex structure, para $=$ paracomplex structure).

p	$\epsilon$	$g_V$	$g_{V^{\perp }}$	J	$J^{\perp }$	$J'$
3	$+$	(2,0)	(2,0)	cx	cx	para
2	$+$	(1,1)	(2,0)	para	cx	cx
2	−	(2,0)	(1,1)	cx	para	cx
1	$+$	(0,2)	(2,0)	cx	cx	para
1	−	(1,1)	(1,1)	para	para	cx
0	−	(0,2)	(1,1)	cx	para	cx

Note also that there is a natural isomorphism $L^{\pm }(S^{p,q} = L^{\mp }(S^{q,p}_-)$, where $S^{q,p}_- = \{ x \in E^{q,p+1}, x^2 =-1 \}$ . We finally reproduce Table 1 in its corrected form.