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Homogeneous almost complex structures in dimension 6 with semi-simple isotropy

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Abstract

We classify invariant almost complex structures on homogeneous manifolds of dimension 6 with semi-simple isotropy. Those with non-degenerate Nijenhuis tensor have the automorphism group of dimension either 14 or 9. An invariant almost complex structure with semi-simple isotropy is necessarily either of specified 6 homogeneous types or a left-invariant structure on a Lie group. For integrable invariant almost complex structures we classify all compatible invariant Hermitian structures on these homogeneous manifolds, indicate their integrability properties (Kähler, SNK, SKT) and mark the other interesting geometric properties (including the Gray-Hervella type).

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Notes

  1. We denote the compact real form of \(G_2\) by \(G_2^c\subset SO(7)\) and the split real form with the trivial center by \(G_2^*\subset SO(3,4)\).

  2. Endomorphisms are \({\mathbb {R}}\)-linear and \(\mathfrak {h}\)-equivariant transformations of the module.

  3. Here and in what follows we adopt the convention that \(x,y\in V\) are arbitrary elements, but \( e \in \mathbb {C}\) is a fixed element, in particular \( e ,i e \) is a real basis of \(\mathbb {C}\).

  4. The central (“left”) extension and extension by derivations (“right”) \(\tilde{\mathfrak {g}}\) of the Lie algebra \(\mathfrak {g}\) (via \(\mathfrak {f}\)) are given respectively by the exact sequences

    $$\begin{aligned} 0\rightarrow \mathfrak {f}\rightarrow \tilde{\mathfrak {g}}\rightarrow \mathfrak {g}\rightarrow 0, \qquad 0\rightarrow \mathfrak {g}\rightarrow \tilde{\mathfrak {g}}\rightarrow \mathfrak {f}\rightarrow 0. \end{aligned}$$

    Then \(\mathfrak {g}\) is respectively the quotient/subalgebra of \(\tilde{\mathfrak {g}}\) and its bracket can/cannot change upon the extension.

  5. This means extension by derivations; terminology comes from Fuks [9], and is opposed to left=central extensions. For a Lie algebra \(\mathfrak {g}\) its “right extensions” are enumerated by the cohomology group \(H^1(\mathfrak {g},\mathfrak {g})\) and “left extensions” by \(H^2(\mathfrak {g})\).

  6. There are obvious pseudo-SNK analogs of signature \((2,4)\) of the last two entries in Bitruilles’ list, but we present here only the spaces \(G/H\) with semi-simple \(H\).

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Authors and Affiliations

  1. Masaryk University, Kotlarska 2, 611 37 , Brno, Czech Republic

    D. V. Alekseevsky

  2. Institute for Information Transmission Problems, B. Karetnuj per. 19, 127994 , Moscow, Russia

    D. V. Alekseevsky

  3. Institute of Mathematics and Statistics, University of Tromsø, Tromsø, 90-37, Norway

    B. S. Kruglikov & H. Winther

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  1. D. V. Alekseevsky
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  2. B. S. Kruglikov
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  3. H. Winther
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Correspondence to B. S. Kruglikov.

Appendix: Tables of the structure of \((\mathfrak {g},\mathfrak {h},\mathfrak {m},J,\omega )\)

Appendix: Tables of the structure of \((\mathfrak {g},\mathfrak {h},\mathfrak {m},J,\omega )\)

Here we list the data for Theorems 1, 3 and indicate some integrability properties for the structures. These completely encode all non-flat homogeneous almost complex manifolds with semi-simple isotropy, only excluding the special cases \(G_2^c/SU(3)=S^6\) and \(G_2^*/SU(2,1)=S^{2,4}\).

Imaginary quaternions are generated by anti-commuting \(i,j,k=ij\), which satisfy \(i^2=-1,j^2=-1\), \(k^2=-1\). In Tables 1 and 2 we identify \(\mathfrak {h}={ \mathtt Im }(\mathbb {H})\).

Table 4 isotropy \(\mathfrak {h}=\mathfrak {su}(1,1)\), representation \(\mathfrak {ad}^\mathbb {C}\)
Full size table

Imaginary split quaternions are generated by anti-commuting \(i,j,k=ij\), with \(i^2=-1,j^2=1\), \(k^2=1\). In Tables 3 and 4 we identify \(\mathfrak {h}={ \mathtt Im }(\mathbb {H}_s)\).

The brackets \([\mathfrak {h},\mathfrak {h}]\) and \([\mathfrak {h},\mathfrak {m}]\) are straightforward and are not included into the tables. We include only the non-trivial brackets \([\mathfrak {m},\mathfrak {m}]\).

The almost complex structure \(J\) is indicated in terms of its 2 parameters for Tables 1, 2, 3 and 4 (beware that the parameter \(r\) has different meaning in different tables). \(J\) has no parameters in Tables 5 and 6. In the case of representation \(\mathfrak {ad}^\mathbb {C}\) we use formula (1) for \(J\).

Table 5 isotropy \(\mathfrak {h}=\mathfrak {sl}_2(\mathbb {C})\), representation \(V+\mathbb {C}\)
Full size table
Table 6 isotropy \(\mathfrak {h}=\mathfrak {sl}_2(\mathbb {C})\), representation \(\mathfrak {ad}\)
Full size table

We list only non-trivial values of the Nijenhuis tensor in the minimal amount due to the identity \(N_J(Jx,y)=N_J(x,Jy)=-JN_J(x,y)\).

We write NDG to indicate that the tensor \(N_J\) is non-degenerate; DG\(_2\) indicates that the image of \(N_J:\Lambda ^2_\mathbb {C}TM\rightarrow TM\) is a real rank 2 subdistribution of \(TM\), DG\(_1\) means it is a real rank 4 subdistribution (in DG\(_k\) the number \(k\) is the complex codimension of \(\mathrm{Im}(N_J)\), see [16]).

The Hermitian metric is given by \(g(\xi ,\eta )=\omega (\xi ,J\eta )\) as in Section 5. Thus we describe only the almost symplectic form \(\omega \) and its differential.

Recall that \(V\) stands for the standard representation, \(\mathbb {C}\) for the trivial complex and \(\mathfrak {ad}\) for the adjoint representation.

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Alekseevsky, D.V., Kruglikov, B.S. & Winther, H. Homogeneous almost complex structures in dimension 6 with semi-simple isotropy. Ann Glob Anal Geom 46, 361–387 (2014). https://doi.org/10.1007/s10455-014-9428-y

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