9 problems

Kobayashi's conjecture. The homogeneous space

Zeghib's conjecture. All compact quotients are standard.

Kobayashi–Yoshino conjecture. A symmetric space admits compact quotients if and only if it admits a standard one.

Compact quotient implies local fibration conjecture. If $G/H$ admits compact quotients, then it admits local geometric fibrations.

Precise geometric fibration conjecture. There exists a $\Gamma$-invariant contractible smooth submanifold $\widetilde{M}$ of $G/K$, of dimension $\dim_+(G/H)$, such that

Locally homogeneous quotient conjecture. Any closed manifold locally modelled on a reductive homogeneous space is a compact quotient of that space.

Compact quotient–local fibration conjecture. The following are equivalent:

Geometric fibration conjecture. The group $\Gamma$ is the fundamental group of a closed aspherical manifold $M$, and there exists a smooth $\Gamma$-equivariant fiber bundle

Space Form Conjecture. There exists a compact, complete pseudo-Riemannian manifold of signature $(p,q)$ with constant sectional curvature $1$ if and only if $(p,q)$ lies in the fol…