Kodai Mathematical Journal

On the canonical Hermitian connection in nearly Kähler manifolds

Luigi Vezzoni

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In the present paper 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, ·, ·)=0 and that it satisfies the first Bianchi identity $\mathfrak{S}\tilde{R}$(X, Y, Z, ·)=0 if and only if g is a Kähler metric. Furthermore we characterize 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.

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Kodai Math. J., Volume 32, Number 3 (2009), 420-431.

First available in Project Euclid: 11 November 2009

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Vezzoni, Luigi. On the canonical Hermitian connection in nearly Kähler manifolds. Kodai Math. J. 32 (2009), no. 3, 420--431. doi:10.2996/kmj/1257948887. https://projecteuclid.org/euclid.kmj/1257948887

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