22 problems

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…

Indefinite energy conjecture. The quantity $Q(\operatorname{\stackrel{\longrightarrow}{Vol}}N)$ is uniquely minimized when $N$ is the intersection of $Q$ with an $a$-plane in $V$ c…

Let $(M,\mathcal{C})$ be a transverse Riemann-Lorentz conformal manifold of dimension $m\geq 4$, with singular set $\Sigma$. Around each singular point $p\in\Sigma$, consider a coo…

Let $D\subset \mathcal V$ be a simply connected domain of an $n$-dimensional $k$-recurrent Lorentzian manifold $(\mathcal V,g)$. A vector field is null recurrent if it is null and…

Let $D\subset \mathcal V$ be a simply connected domain of an $n$-dimensional $k$-symmetric Lorentzian manifold $(\mathcal V,g)$. A vector field is null covariantly constant if it i…

Auslander conjecture, special case. The fundamental group $\pi_1(M)$ contains a solvable subgroup of finite index.

Generalized Calabi–Markus conjecture. The fundamental group $\pi_1(M)$ is finite, and $M$ is noncompact.

Conjecture on negative transverse curvature. There is a Riemannian metric $g$ on $M$ such that the function $\chi_{g,\bot H}$ is everywhere negative.

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…

Standard quotient conjecture. The homogeneous space $G/H$ admits a cocompact properly discontinuous group if and only if $G/H$ admits a compact standard quotient.

Let $(M,g)$ be a pseudo-Riemannian manifold. Its conformal group is denoted by $\mathrm{Conf}^\uparrow(M,g)$, and this group is essential if there is no metric conformal to $g$ for…

Reductive constructor conjecture. If the homogeneous space $G/H$ of reductive type admits compact Clifford–Klein forms, that is, $G/H$ admits a reductive constructor.

The conjecture. The only compact locally homogeneous simple pseudo-Riemannian Einstein manifolds are of the form $\Gamma\backslash G/H$, determined by either some standard triple…

Let $(\mathfrak{g}^{\mathbb{C}}, g^{\mathbb{C}})$ be a holomorphic inner product space, with $\mathfrak{g}^{\mathbb{C}}$ a complex Lie algebra. Let…

Transitivity conjecture. If $M$ is an $n$-dimensional Lorentzian space with all curvature invariants constant, then there exists a transitive set $\mathcal{N}$.

Generalized Lichnerowicz conjecture. If a compact Lorentzian manifold has an essential conformal group, then it is conformally flat.

Let $({\cal M},\mathrm{g})$ be an indecomposable locally homogeneous pp-wave of dimension greater than $3$. A locally homogeneous pp-wave is a pp-wave whose local Killing vector fi…

Dušek's $k=0$ conjecture. For all null homogeneous geodesics in $G/H$, one has $k=0$ in the geodesic lemma.

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 a pseudo-Riemannian space form of signature $(p,q)$ be a pseudo-Riemannian manifold with constant sectional curvature $\kappa$, where $p,q\neq 0$. Kobayashi's pseudo-Riemannian…

Finiteness conjecture. Unless $(M,g)$ is a finite quotient of the standard Riemannian sphere,

Consider the double-coset $G$-spaces of the form $K\backslash L/\Gamma$, where $L$ is a semisimple Lie group with an irreducible lattice $\Gamma$ and a compact subgroup $K$, and th…