1 Introduction

Correction to: Abh. Math. Semin. Univ. Hambg. (2022) 92:105–123 https://doi.org/10.1007/s12188-021-00254-y

Let \(E^4_1\) be the Minkowski 4-space and \(\bigwedge ^2 E^4_1\) the 2-fold exterior power of \(E^4_1\). Then \(\bigwedge ^2 E^4_1\) is of dimension 6 and the Minkowski metric of \(E^4_1\) induces an indefinite metric of \(\bigwedge ^2 E^4_1\) with signature (3, 3). The SO(3, 1)-action on \(E^4_1\) yields an SO(3, 1)-action on \(\bigwedge ^2 E^4_1\). In addition, each element of SO(3, 1) gives an isometry of \(\bigwedge ^2 E^4_1\). In particular, we have an SO(3, 1)-action on the light cone \(\mathcal {L}\) of \(\bigwedge ^2 E^4_1\). In the paragraph just before [2, Proposition 1], two hypersurfaces \(\mathcal {L}_{\pm }\) of \(\mathcal {L}\) are given. These are SO(3, 1)-orbits in \(\mathcal {L}\). In this proposition, it was asserted that \(\mathcal {L}_{\pm }\) are neutral, that is, they have neutral metrics. This assertion has no problems. However, we will see in this paper that \(\mathcal {L}_{\pm }\) are not flat, although it was asserted that \(\mathcal {L}_{\pm }\) are flat in [2, Proposition 1]. By the equation of Gauss for submanifolds \(\mathcal {L}_{\pm }\) of \(\bigwedge ^2 E^4_1\), we can explicitly represent the curvature tensors of \(\mathcal {L}_{\pm }\), and we will see that they do not vanish. In [2, Proposition 1], it was also asserted that \(\mathcal {L}_{\pm }\) are neutral hyperKähler. However, according to the proof of [2, Proposition 1], this assertion is based on the flatness. Therefore the assertion that \(\mathcal {L}_{\pm }\) are neutral hyperKähler must be cancelled. Hence we see that Proposition 1 of [2] should be stated as follows:

The 4-submanifolds \(\mathcal {L}_{\pm }\) are neutral and not flat.

In this paper, we will find one parallel almost complex structure and one parallel almost paracomplex structure on each of \(\mathcal {L}_{\pm }\). In addition, we will see that they are suitable for Theorems 1 and 2 in [2]. Therefore these theorems have no problems.

2 The curvature tensors

As was used in the proof of [2, Proposition 1], let \(\tilde{\nabla }^+\) be the Levi-Civita connection of the metric of \(\mathcal {L}_+\) induced by the metric \(\hat{h}\) of \(\bigwedge ^2 E^4_1\) and S a surface in \(\mathcal {L}_+\) given by \(S=\{ \tilde{T}_{P_{3, 1}} \circ \tilde{T}_{P_{3, 2}} (E_{+, 1} ) \ | \ \theta , t\in {\varvec{R}} \}\), where \(E_{\pm , i}\) (\(i=1, 2, 3\)) are given in the second paragraph of [2, Section 2] and \(\tilde{T}_{P_{k, l}}\) (\(k=1, 2, 3\), \(l=1, 2\)) are given in the proof of [2, Proposition 1]. Then vector fields \(E'_{\pm , 2}\), \(E'_{\pm , 3}\) along S given in the proof of [2, Proposition 1] are parallel with respect to \(\tilde{\nabla }^+\). Let \(\hat{\nabla }\) be the Levi-Civita connection of \(\hat{h}\). Then \(E'_{\pm , 3}\) are parallel with respect to \(\hat{\nabla }\), while \(E'_{\pm , 2}\) are not parallel with respect to \(\hat{\nabla }\). Let \(\omega _{ij}\) be as in the second paragraph of [2, Section 2]. Then \(\omega _{13}\), \(\omega _{42}\), \(\omega _{23}\), \(\omega _{14}\) form a pseudo-orthonormal basis of the tangent space of \(\mathcal {L}_+\) at a point \(E_{+, 1}\). In addition, \(\omega _{13}\), \(\omega _{42}\) form a pseudo-orthonormal basis of the tangent plane of S at the same point. Let \(\omega '_{ij}\) be vector fields along S given by \(\omega '_{ij} =\tilde{T}_{P_{3, 1}} \circ \tilde{T}_{P_{3, 2}} (\omega _{ij} )\). Then using

$$\begin{aligned} \begin{aligned} \tilde{T}_{P_{3, 1}} (E_{\pm , 2} )&=-\sin \theta E_{\pm , 1} +\cos \theta E_{\pm , 2} , \\ \tilde{T}_{P_{3, 2}} (E_{\mp , 2} )&= \mp \textrm{sinh}\,t E_{\pm , 1} +\textrm{cosh}\,t E_{\mp , 2} , \end{aligned} \end{aligned}$$

which were already obtained in the proof of [2, Proposition 1], we obtain

$$\begin{aligned} \begin{aligned}&\hat{\nabla }_{\omega _{13}} \omega '_{13} =-\hat{\nabla }_{\omega _{42}} \omega '_{42} =-\dfrac{1}{\sqrt{2}} (\omega _{12} -\omega _{34} ), \\&\hat{\nabla }_{\omega _{13}} \omega '_{42} = \hat{\nabla }_{\omega _{42}} \omega '_{13} =-\dfrac{1}{\sqrt{2}} (\omega _{12} +\omega _{34} ). \end{aligned} \end{aligned}$$
(1)

Referring to the previous paragraph, we have an analogous study along a surface \(S^{\perp }\) in \(\mathcal {L}_+\) given by \(S^{\perp } =\{ \tilde{T}_{P_{2, 1}} \circ \tilde{T}_{P_{2, 2}} (E_{+, 1} ) \ | \ \theta , t\in {\varvec{R}} \}\). Then \(\omega _{23}\), \(\omega _{14}\) form a pseudo-orthonormal basis of the tangent plane of \(S^{\perp }\) at \(E_{+, 1}\). Let \(\omega ''_{ij}\) be vector fields along \(S^{\perp }\) given by \(\omega ''_{ij} =\tilde{T}_{P_{2, 1}} \circ \tilde{T}_{P_{2, 2}} (\omega _{ij} )\). Then we obtain

$$\begin{aligned} \begin{aligned}&\hat{\nabla }_{\omega _{23}} \omega ''_{23} =-\hat{\nabla }_{\omega _{14}} \omega ''_{14} =-\dfrac{1}{\sqrt{2}} (\omega _{12} -\omega _{34} ), \\&\hat{\nabla }_{\omega _{14}} \omega ''_{23} = \hat{\nabla }_{\omega _{23}} \omega ''_{14} =-\dfrac{1}{\sqrt{2}} (\omega _{12} +\omega _{34} ). \end{aligned} \end{aligned}$$
(2)

Let \(\tilde{R}^+\), \(\hat{R}\) be the curvature tensors of \(\tilde{\nabla }^+\), \(\hat{\nabla }\) respectively. Then using (1), (2) and the equation of Gauss for \(\mathcal {L}_+\):

$$\begin{aligned} \begin{aligned} 0&=\hat{h} (\hat{R} (X, Y)Z, W) \\&=\hat{h} (\tilde{R}^+ (X, Y)Z, W) +\hat{h} (\sigma (X, Z), \sigma (Y, W)) -\hat{h} (\sigma (X, W), \sigma (Y, Z)) \end{aligned} \end{aligned}$$

(\(\sigma \) is the second fundamental form of \(\mathcal {L}_+\) in \(\bigwedge ^2 E^4_1\)), we obtain

Proposition 1

If we set

$$\begin{aligned} A=\left[ \begin{array}{cccc} 0 &{} 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} -1 &{} 0 \\ 0 &{} -1 &{} 0 &{} 0 \\ 1 &{} 0 &{} 0 &{} 0 \end{array} \right] , \quad B=\left[ \begin{array}{cccc} 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \\ -1 &{} 0 &{} 0 &{} 0 \\ 0 &{} -1 &{} 0 &{} 0 \end{array} \right] , \end{aligned}$$

then the following hold : 

$$\begin{aligned} \begin{aligned}&\tilde{R}^+ (\omega _{13} , \omega _{42} )=0, \\&\tilde{R}^+ (\omega _{23} , \omega _{14} )=0, \\&(\tilde{R}^+ (\omega _{13} , \omega _{23} )\omega _{13} \ \tilde{R}^+ (\omega _{13} , \omega _{23} )\omega _{42} \ \tilde{R}^+ (\omega _{13} , \omega _{23} )\omega _{23} \ \tilde{R}^+ (\omega _{13} , \omega _{23} )\omega _{14} ) \\&\quad =-(\tilde{R}^+ (\omega _{42} , \omega _{14} )\omega _{13} \ \tilde{R}^+ (\omega _{42} , \omega _{14} )\omega _{42} \ \tilde{R}^+ (\omega _{42} , \omega _{14} )\omega _{23} \ \tilde{R}^+ (\omega _{42} , \omega _{14} )\omega _{14} ) \\&\quad = (\omega _{13} \ \omega _{42} \ \omega _{23} \ \omega _{14} )A, \\&(\tilde{R}^+ (\omega _{13} , \omega _{14} )\omega _{13} \ \tilde{R}^+ (\omega _{13} , \omega _{14} )\omega _{42} \ \tilde{R}^+ (\omega _{13} , \omega _{14} )\omega _{23} \ \tilde{R}^+ (\omega _{13} , \omega _{14} )\omega _{14} ) \\&\quad = (\tilde{R}^+ (\omega _{42} , \omega _{23} )\omega _{13} \ \tilde{R}^+ (\omega _{42} , \omega _{23} )\omega _{42} \ \tilde{R}^+ (\omega _{42} , \omega _{23} )\omega _{23} \ \tilde{R}^+ (\omega _{42} , \omega _{23} )\omega _{14} ) \\&\quad = (\omega _{13} \ \omega _{42} \ \omega _{23} \ \omega _{14} )B. \end{aligned} \end{aligned}$$

From Proposition 1, we see that \(\mathcal {L}_+\) is not flat. Similarly, we see that \(\mathcal {L}_-\) is not flat.

3 Complex structures and paracomplex structures

Let \(\mathcal {T}_{E_{+, 1}} (\mathcal {L}_+ )\) denote the tangent space of \(\mathcal {L}_+\) at a point \(E_{+, 1}\). Let \(\hat{\wedge }\) denote the exterior product of the exterior algebra of \(\mathcal {T}_{E_{+, 1}} (\mathcal {L}_+ )\). Then we denote by \(\hat{\bigwedge }^2 \mathcal {T}_{E_{+, 1}} (\mathcal {L}_+ )\) the 2-fold exterior power of \(\mathcal {T}_{E_{+, 1}} (\mathcal {L}_+ )\). We set

$$\begin{aligned} X_1:=\omega _{23}, \quad X_2:=\omega _{14}, \quad Y_1:=\omega _{13}, \quad Y_2:=\omega _{42}. \end{aligned}$$

Then \(\hat{\bigwedge }^2 \mathcal {T}_{E_{+, 1}} (\mathcal {L}_+ )\) is decomposed into

$$\begin{aligned} \hat{\bigwedge }^2 \mathcal {T}_{E_{+, 1}} (\mathcal {L}_+ ) =\hat{\bigwedge }^2_+ \mathcal {T}_{E_{+, 1}} (\mathcal {L}_+ ) \oplus \hat{\bigwedge }^2_- \mathcal {T}_{E_{+, 1}} (\mathcal {L}_+ ), \end{aligned}$$

where

  1. (i)

    \(\hat{\bigwedge }^2_+ \mathcal {T}_{E_{+, 1}} (\mathcal {L}_+ )\) is generated by

    $$\begin{aligned} \dfrac{1}{\sqrt{2}} (X_1 \hat{\wedge } Y_1 -X_2 \hat{\wedge } Y_2 ), \ \dfrac{1}{\sqrt{2}} (X_1 \hat{\wedge } X_2 +Y_2 \hat{\wedge } Y_1 ), \ \dfrac{1}{\sqrt{2}} (X_1 \hat{\wedge } Y_2 +Y_1 \hat{\wedge } X_2 ), \end{aligned}$$
  2. (ii)

    \(\hat{\bigwedge }^2_- \mathcal {T}_{E_{+, 1}} (\mathcal {L}_+ )\) is generated by

    $$\begin{aligned} \dfrac{1}{\sqrt{2}} (X_1 \hat{\wedge } Y_1 +X_2 \hat{\wedge } Y_2 ), \ \dfrac{1}{\sqrt{2}} (X_1 \hat{\wedge } X_2 -Y_2 \hat{\wedge } Y_1 ), \ \dfrac{1}{\sqrt{2}} (X_1 \hat{\wedge } Y_2 -Y_1 \hat{\wedge } X_2 ). \end{aligned}$$

The stabilizer \(G(E_{+, 1} )\) of SO(3, 1) at \(E_{+, 1}\) is generated by \(P_{1, 1}\), \(\pm P_{1, 2}\) (\(\theta \), \(t\in {\varvec{R}}\)). Then \(G(E_{+, 1} )\) acts on \(\mathcal {T}_{E_{+, 1}} (\mathcal {L}_+ )\). Therefore \(G(E_{+, 1} )\) acts on \(\hat{\bigwedge }^2 \mathcal {T}_{E_{+, 1}} (\mathcal {L}_+ )\).

We see that \((1/\sqrt{2} )(X_1 \hat{\wedge } Y_1 -X_2 \hat{\wedge } Y_2 )\) is an invariant element of \(\hat{\bigwedge }^2_+ \mathcal {T}_{E_{+, 1}} (\mathcal {L}_+ )\) by the \(G(E_{+, 1} )\)-action, which is unique up to a constant, and \((1/\sqrt{2} )(X_1 \hat{\wedge } Y_1 -X_2 \hat{\wedge } Y_2 )\) defines an almost complex structure \(\mathcal {I}_+\) on \(\mathcal {L}_+\) by the SO(3, 1)-action. Using (1) and (2), and referring to [1], we see that \(\mathcal {I}_+\) is parallel with respect to \(\tilde{\nabla }^+\).

We see that \((1/\sqrt{2} )(X_1 \hat{\wedge } Y_2 -Y_1 \hat{\wedge } X_2 )\) is an invariant element of \(\hat{\bigwedge }^2_- \mathcal {T}_{E_{+, 1}} (\mathcal {L}_+ )\) by the \(G(E_{+, 1} )\)-action, which is unique up to a constant, and \(-(1/\sqrt{2} )(X_1 \hat{\wedge } Y_2 -Y_1 \hat{\wedge } X_2 )\) defines an almost paracomplex structure \(\mathcal {J}_+\) on \(\mathcal {L}_+\) by the SO(3, 1)-action. Using (1) and (2), and referring to [1], we see that \(\mathcal {J}_+\) is parallel with respect to \(\tilde{\nabla }^+\).

We have similar discussions for \(\mathcal {L}_-\) and we obtain an almost complex structure \(\mathcal {I}_-\) and an almost paracomplex structure \(\mathcal {J}_-\) on \(\mathcal {L}_-\), which are parallel with respect to the Levi-Civita connection \(\tilde{\nabla }^-\) of the metric of \(\mathcal {L}_-\) induced by \(\hat{h}\). Hence we obtain

Proposition 2

For \(\varepsilon \in \{ +, -\}\), \(\mathcal {L}_{\varepsilon }\) has just two almost complex structures \(\pm \mathcal {I}_{\varepsilon }\) and just two almost paracomplex structures \(\pm \mathcal {J}_{\varepsilon }\) by the SO(3, 1)-action and these are parallel with respect to \(\tilde{\nabla }^{\varepsilon }\).

We see that \(\mathcal {I}_{\pm }\), \(\mathcal {J}_{\pm }\) satisfy (5), (6) in the proof of [2, Proposition 1] respectively. Therefore Theorems 1 and 2 have no problems.