Curvatures of real connections on Hermitian manifolds

Abstract

Let $(M,g,J)$ be a Riemannian manifold with a compatible integrable complex structure $J\in \mathrm{End}(T_{\mathbb{R}}M)$ and ${\mathcal{A}}_{g,J}$ be the space of real connections on $T_{\mathbb{R}}M$ preserving both $g$ and $J$. In this paper, we investigate the relationship between the geometry of real connections in ${\mathcal{A}}_{g,J}$ and that of Hermitian connections on $T^{1,0}M$. In particular, we study the geometry of the real Chern connection ${\nabla }^{\mathrm{Ch},\mathrm{R}}$ on $(M,g,J)$, and obtain Kähler-Einstein metrics by using real Chern-Einstein metrics.

References in corpus (4)

• Non-Kaehler Heterotic String Compactifications with non-zero fluxes and constant dilaton
• Kodaira dimensions of almost complex manifolds II
• Scalar curvature, Kodaira dimension and $\hat{A}$-genus
• Chern-Ricci curvatures, holomorphic sectional curvature and Hermitian metrics