5 problems

Fujiki-manifold conjecture. If a Fujiki manifold admits a holomorphic Cartan geometry, then it admits a flat holomorphic Cartan geometry with the same model, and the Fujiki manifol…

Let $G$ be the split real form of a semisimple Lie group with parabolic subgroup $P$, and let $P\to E\to M$ be a $G/P$-geometry. Suppose that the Lie algebra $\mathfrak{g}$ has its…

Let $S$ and $S'$ be Riemannian surfaces whose Gauss curvatures differ at every pair of points, and let $M$ be the space of linear isometries between tangent planes of $S$ and $S'$.…

Let an $\mathcal{O}(n)$-geometry be a Cartan geometry modelled on the corresponding model, and call it tame according to the source's definition. Tameness conjecture. An…

Let $M$ be a manifold with a normal projective connection, and suppose that its geodesics are embedded curves. Projective Blaschke conjecture. If all geodesics are closed embedded…