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 K\"ahler-Einstein metrics by using real Chern-Einstein metrics.
Categories: math.DG