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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
which were already obtained in the proof of [2, Proposition 1], we obtain
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
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 \(\mathcal {L}_+\):
(\(\sigma \) is the second fundamental form of \(\mathcal {L}_+\) in \(\bigwedge ^2 E^4_1\)), we obtain
Proposition 1
If we set
then the following hold :
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
Then \(\hat{\bigwedge }^2 \mathcal {T}_{E_{+, 1}} (\mathcal {L}_+ )\) is decomposed into
where
-
(i)
\(\hat{\bigwedge }^2_+ \mathcal {T}_{E_{+, 1}} (\mathcal {L}_+ )\) is generated by
$$\begin{aligned} \dfrac{1}{\sqrt{2}} (X_1 \hat{\wedge } Y_1 -X_2 \hat{\wedge } Y_2 ), \ \dfrac{1}{\sqrt{2}} (X_1 \hat{\wedge } X_2 +Y_2 \hat{\wedge } Y_1 ), \ \dfrac{1}{\sqrt{2}} (X_1 \hat{\wedge } Y_2 +Y_1 \hat{\wedge } X_2 ), \end{aligned}$$ -
(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.
References
Ando, N.: Surfaces with zero mean curvature vector in neutral \(4\)-manifolds. Differ. Geom. Appl. 72, 101647 (2020)
Ando, N.: Isotropicity of surfaces in Lorentzian 4-manifolds with zero mean curvature vector. Abh. Math. Semin. Univ. Hambg. 92, 105–123 (2022)
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The author would like to express his cordial gratitude to the referee for helpful comments.
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Communicated by Vicente Cortés.
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Ando, N. Correction to: Isotropicity of surfaces in Lorentzian 4-manifolds with zero mean curvature vector. Abh. Math. Semin. Univ. Hambg. 93, 163–166 (2023). https://doi.org/10.1007/s12188-023-00272-y
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DOI: https://doi.org/10.1007/s12188-023-00272-y