On the canonical Hermitian connection in nearly Kähler manifolds. (English) Zbl 1180.53071

Vezzoni, Luigi

Kodai Math. J. 32, No. 3, 420-431 (2009).

Summary: We prove that the Hermitian curvature tensor \(\tilde R\) associated to a nearly Kähler metric \(g\) always satisfies the second Bianchi identity \(\mathfrak S(\tilde \nabla _X \tilde R)(Y, Z, \cdot , \cdot )=0\) and that it satisfies the first Bianchi identity \(\mathfrak S \tilde R(X, Y, Z, \cdot )=0\) if and only if \(g\) is a Kähler metric. Furthermore we give a condition for \(\tilde R\) to be parallel with respect to the canonical Hermitian connection \(\tilde \nabla\) in terms of the Riemann curvature tensor and in the last part of the paper we study the curvature of some generalizations of the nearly Kähler structure.

MSC:

53C55

Global differential geometry of Hermitian and Kählerian manifolds

Keywords:

Bianchi identity; curvature tensor; nearly Kähler structure