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

Ganchev, G.; Kassabov, O.

Kodai Math. J. 29, No. 2, 281-298 (2006).

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.

Reviewer: Zbigniew Olszak (Wrocław)

MSC:

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

Keywords:

complex manifold; complex connection; Hermitian manifold; locally conformal Kähler manifold