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On Bismut Flat Manifolds

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arxiv 1603.07058 v3 pith:SL6YKAQP submitted 2016-03-23 math.DG

classification math.DG
keywords bismutflatcompactmanifoldmanifoldsonlysurfacesthreefolds
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In this paper, we give a classification of all compact Hermitian manifolds with flat Bismut connection. We show that the torsion tensor of such a manifold must be parallel, thus the universal cover of such a manifold is a Lie group equipped with a bi-invariant metric and a compatible left invariant complex structure. In particular, isosceles Hopf surfaces are the only Bismut flat compact non-K\"ahler surfaces, while central Calabi-Eckmann threefolds are the only simply-connected compact Bismut flat threefolds.

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  1. On Strominger K\"ahler-like manifolds with degenerate torsion

    math.DG 2019-08 accept novelty 7.0 of 10

    Every complete non-Kähler SKL threefold has universal cover either a product of two Sasakian 3-manifolds or a non-Kähler SKL surface times a Kähler curve, and degenerate-torsion SKL manifolds split off a Kähler factor.

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