Abstract
In [2], Borisov, Salamon and Viaclovsky constructed non-standard orthogonal complex structures on flat tori TR2n for any n≥3. We will call these examples BSV-tori. In this note, we show that on a flat 6-torus, all the orthogonal complex structures are either the complex tori or the BSV-tori. This solves the classification problem for compact Hermitian manifolds with flat Riemannian connection in the case of complex dimension three.
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