12 problems

Let $M^{3}$ be a closed, prime, atoroidal $3$-manifold. A locally homogeneous Riemannian metric is a Riemannian metric such that for any two points $p$ and $q$ there is an isometry…

Let $S$ be the closed surface underlying the quasi-Hitchin representations under consideration, let $M_{\mathbb C}$ be the manifold associated with a quasi-Hitchin representation i…

Local homogeneity conjecture. Any such structure $\phi$ is locally homogeneous. This implies that if $\phi$ is rigid, then the fundamental group of $X$ is infinite.

Local homogeneity conjecture. Any holomorphic Riemannian metric on $X$ is locally homogeneous.

Guichard-Wienhard conjecture. There exists a generalized flag variety $X$ and a compact fiber bundle $M\to S$ such that $\overline{\rho}$ is the holonomy of a locally homogeneous…

Soit $M$ une variété compacte munie d'une structure affine, c'est-à-dire d'une structure modelée sur $(\operatorname{Aff}(\mathbb{R}^n),\mathbb{R}^n)$. Une forme volume parallèle e…

Let $M$ be a compact conformally flat Lorentzian manifold with developing map $\operatorname{dev}:\widetilde M\to \mathbb{R}^{2n+2}$, and suppose that $\operatorname{dev}$ is not s…

Let $M$ be a complex manifold and let $S \subset M$ be a subset of complex codimension at least two. A holomorphic Engel $2$-plane field is a holomorphic $2$-plane field satisfying…

Let $M$ be a domain in a Stein manifold, let $\hat{M}$ be its envelope of holomorphy, and let a holomorphic first order structure on $M$ have an underlying holomorphic principal bu…

Let $D$ be a semisimple Lie group all of whose factors have property $(T)$, or a lattice in such a Lie group. A rigid geometric structure is a geometric structure of the rigid type…

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 compact complex manifold endowed with a holomorphic Riemannian metric of constant non-vanishing curvature. Completeness conjecture. The metric on $M$ is complete; in p…