Kodai Mathematical Journal

On the canonical Hermitian connection in nearly Kähler manifolds

Luigi Vezzoni

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Abstract

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.

Article information

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

Dates
First available in Project Euclid: 11 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.kmj/1257948887

Digital Object Identifier
doi:10.2996/kmj/1257948887

Mathematical Reviews number (MathSciNet)
MR2582009

Zentralblatt MATH identifier
1180.53071

Citation

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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