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5 votes
0 answers
129 views

Complete Hermitian manifolds with vanishing Chern curvature

An old theorem going back to Boothby states that a compact Hermitian manifold with Chern curvature vanishing identically is a compact quotient of a complex Lie group with a left invariant metric. Are ...
4 votes
1 answer
588 views

Affine space structure on the space of Hermitian connections

I'm reading Gauduchon's paper Hermitian connections and Dirac operators. For a fixed almost-Hermitian manifold $(M, g, J)$ let $\mathcal A(g, J)$ be the space of connections $\nabla$ s.t. $\nabla g = ...
1 vote
1 answer
210 views

Hermitic connections on complex line bundles with imaginary curvature form

It is a simple fact that if $L \to B$ is a complex line bundle endowed with an Hermitian product and a compatible connection $\nabla$, then the curvature $F_\nabla$ is imaginary (and so are the local ...