Abstract
We classify flat strict nearly Kähler manifolds with (necessarily) indefinite metric. Any such manifold is locally the product of a flat pseudo-Kähler factor of maximal dimension and a strict flat nearly Kähler manifold of split signature (2m, 2m) with m ≥ 3. Moreover, the geometry of the second factor is encoded in a complex three-form \(\zeta \in \Lambda^3 (\mathbb{C}^m)^*\). The first nontrivial example occurs in dimension 4m = 12.
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Cortés, V., Schäfer, L. Flat nearly Kähler manifolds. Ann Glob Anal Geom 32, 379–389 (2007). https://doi.org/10.1007/s10455-007-9068-6
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DOI: https://doi.org/10.1007/s10455-007-9068-6
Keywords
- Nearly Kähler manifolds
- Flat almost pseudo-Hermitian manifolds
- Pseudo-Riemannian manifolds
- Almost complex structures