194 problems

A homogeneous expanding Ricci soliton is a homogeneous Riemannian manifold whose Ricci flow evolves by scaling and diffeomorphisms with an expanding scale factor. Generalized Aleks…

Kobayashi's conjecture. The homogeneous space

Bing–Borsuk conjecture. Every $n$-dimensional homogeneous metric ANR-compactum is an $n$-manifold.

Let $\mathcal{M}$ be a homogeneous quaternionic Kähler manifold with negative Ricci curvature. The manifold is called normal if it admits a transitive solvable group of isometries…

Homogeneity Conjecture. The manifold $(M',ds'^2)$ is homogeneous if and only if every $\gamma\in\Gamma$ is an isometry of constant displacement on $(M,ds^2)$.

Let $\mathrm{G}$ be a semisimple algebraic group and let $\mathrm{P}\subset \mathrm{G}$ be a parabolic subgroup. Let ${\mathbf D^{\mathrm{b}}}(\mathrm{G}/\mathrm{P})$ denote the bo…

Let $G$ be a compact connected Lie group. For a left-invariant Riemannian metric $g$ on $G$, let $\lambda_1(G,g)$ denote the smallest positive eigenvalue of the Laplace–Beltrami op…

A noncompact Einstein homogeneous space is a noncompact homogeneous Riemannian manifold whose metric is Einstein, and an isometry group acts simply transitively when through every…

Let $Z$ be a complex projective manifold equipped with a contact structure, namely a holomorphic hyperplane distribution $D\subset TZ$ such that the induced bundle morphism … is ev…

Modified Bing–Borsuk conjecture. Every locally compact homogeneous $ANR$-space of dimension $n\geq 3$ is a generalized $n$-manifold.

de Groot's conjecture. Every $n$-dimensional absolute suspension is homeomorphic to the $n$-sphere.

Holonomic rank conjecture. The solution rank of $\tau$ at the point $a\in V^\vee$ is given by

Let a homogeneous space, such as a Grassmannian manifold, and simpler homogeneous spaces, such as products of projective spaces, have associated hypergeometric series; twisted hype…

Serre's conjecture II. Every principal homogeneous space under a semisimple simply connected $K$-group has a rational point.

Gray–Wolf converse conjecture. Every homogeneous nearly Kähler manifold is a 3-symmetric space.

Homogeneity Conjecture. The quotient $\Gamma\backslash M$ is homogeneous if and only if every $\gamma\in\Gamma$ is an isometry of constant displacement on $M$.

Finiteness conjecture. This set is finite.

Connection-blocking conjecture. The space $M$ is blockable if and only if

Let $G/H$ be a compact homogeneous space whose isotropy representation consists of pairwise inequivalent irreducible summands; in particular, this includes the case where…

An isoparametric hypersurface in a sphere is a hypersurface with constant principal curvatures; let $g$ denote the number of distinct principal curvatures. Dorfmeister–Neher conjec…

Gray's conjecture. Any homogeneous nearly Kähler manifold must be 3-symmetric.

Let $X$ be a homogeneous space, and let $QH^*(X,\mathbb{Z})_{loc}$ denote its quantum cohomology ring localized by inverting the quantum parameter. The homogeneous space $X$ is cal…

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

Let $\mathbb{S}_q$ denote the isotropic Grassmannian appearing in the surrounding discussion. Isotropic-Grassmannian map conjecture. If $p<q$, then there are no nonconstant maps ……

Naive $k$-ampleness conjecture. The vector bundle ${\cal E}$ is $k_X$-ample.