Hermitian manifolds with flat associated connection. (English) Zbl 1117.53052

Let \(M(J)\) be a complex manifold with complex structure \(J\). For a symmetric affine connection \(\nabla\) on \(M\), there exists a unique symmetric complex connection \(\widetilde\nabla\) such that \(\widetilde\nabla_XY=(\nabla_XY)^{1,0}\) for any vector fields \(X,Y\) of type \((1,0)\). \(\widetilde\nabla\) is said to be associated to \(\nabla\). Let additionally \(M(J,g)\) be Hermitian and \(\nabla\) be its Levi-Civita connection. When the associated connection \(\widetilde\nabla\) is flat, the manifold admits locally special holomorphic coordinates in which the components of \(\widetilde\nabla\) are zero, and therefore in this case a local classification of the Hermitian metric \(g\) is obtained.
Any conformal transformation of \(g\) induces locally a holomorphically projective transformation (with closed \(1\)-form) of \(\widetilde\nabla\) and vice versa. Then the holomorphically projective tensor of \(\tilde\nabla\) gives rise to an associated conformal curvature tensor \(\widetilde W\) which is a conformal invariant. A geometric interpretation of the vanishing of \(\widetilde W\) is given in the class of locally conformal Kählerian manifolds.


53C55 Global differential geometry of Hermitian and Kählerian manifolds
53B35 Local differential geometry of Hermitian and Kählerian structures
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