5 problems

Podestà–Zuddas conjecture. Every $G$-invariant BTP metric on $(G/H,J)$ is either a Kähler metric or a multiple of the standard metric $g_o$.

Bismut space-form conjecture. If a compact Hermitian manifold $(M^n,g)$ has constant Bismut holomorphic sectional curvature $c$ and $c\neq 0$, then $g$ must be Kähler.

Chen–Zheng conjecture. If the holomorphic sectional curvature of $\nabla^b$ is a non-zero constant, then $g$ must be Kähler, hence a complex space form.

BTP-to-BAS conjecture. Under these assumptions, $g$ must be BAS.

Let $(M^n,g)$ be a compact (or complete) Hermitian manifold. Its Bismut connection is denoted by $\nabla^b$, and a connection is Ambrose–Singer when its torsion and curvature are p…