101 problems

Bryant's modified Bing–Borsuk conjecture. Every locally compact homogeneous ANR-space of dimension $n\geq 3$ is a generalized $n$-manifold. This modifies the Bing–Borsuk conjecture…

Let $G/H$ be a homogeneous space in the reductive setting, and let $K$ be a maximal compact subgroup of $G$. Rank inequality conjecture. If $G/H$ admits a cocompact discontinuous g…

A pseudo-Riemannian manifold of signature $(p,q)$ is understood here to be compact and complete, with constant sectional curvature $1$. Space form conjecture. There exists such a m…

Nonexistence conjecture for $\operatorname{SO}(2,2m+1)/\operatorname{SU}(1,m)$. The homogeneous space

Gray and Wolf's conjecture. Every nearly Kähler homogeneous manifold is a 3-symmetric space equipped with its canonical almost complex structure.

Converse to Theorem 15. The converse of Theorem 15 also holds.

Let $G$ be a semisimple algebraic group and let $G/P$ be a projective homogeneous space of $G$. A collection of vector bundles on $G/P$ is exceptional if its derived endomorphism c…

Let $Y$ be a Fano complex contact manifold with second Betti number $b_2=1$. For a simple Lie group $G$, let $\mathfrak{g}$ denote its Lie algebra, and let $G$ act on the projectiv…

Gray–Wolf conjecture. Every homogeneous strictly nearly Kähler space is a naturally reductive 3-symmetric space equipped with its canonical almost-complex structure.

Let $G/P$ be a homogeneous space, let $M_\lambda$ be the maximal proper $P$-submodule associated with a parameter $\lambda$, and let…

Let $G$ be a Lie group, let $H$ be a closed subgroup, and let $d(H)$ denote the invariant used in the paper to measure the dimension relevant to tessellations of $G/H$. One-dimensi…

Nonexistence conjecture. None of these homogeneous spaces has a tessellation. These are presented as three special cases of Kobayashi's general conjecture; the supplied source give…

Let $F(n)$ be a full flag manifold with $n\geq 5$, equipped with a complex structure $J$ and its associated tournament ${\cal T}(J)$. Let $ds^2_{\Lambda}$ denote the Borel metric.…

Let $G$ be an algebraic group and $H\subseteq G$ an observable subgroup. Write $W(H)=N_G(H)/H$ for the corresponding quotient of the normalizer. The subgroup $H$ is reductive if it…

Cayley hypersurface characterisation conjecture. Then, in a suitable affine coordinate system, $\Sigma$ is given by the Cayley hypersurface equation.

Let $G$ be a compact Lie group, let $H$ be a subgroup, and let $\mathfrak v$ denote the corresponding tangent-space representation used to form the invariant polynomial algebra…

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

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

Kobayashi's conjecture. The reductive homogeneous space $G/H$ admits compact quotients if and only if it admits standard ones.

Folklore conjecture. Then $(X,g)$ is a homogeneous space.

Three-dimensional classification conjecture. If $(X,g)$ is a $3$-dimensional simply connected manifold with invariant horospherical mean curvature function, then $X$ is, up to scal…

Hodge–Tate and semisimplicity conjecture. The following should hold:

Dynamical Alekseevskii conjecture. If a homogeneous space has an immortal Ricci flow, then its universal cover is diffeomorphic to $\mathbb{R}^n$.