43 problems

Let $M$ be a closed affine manifold whose linear holonomy lies in $\operatorname{SL}_d(\mathbb{R})$, so that $M$ has parallel volume. An affine manifold is complete if it is a quot…

Let a Lorentzian conformal manifold be a manifold equipped with a conformal class of Lorentzian metrics, and let $\operatorname{Ein}^{1,n}$ denote the Lorentzian Einstein space. Tw…

Let $(M^n,g)$ be a pseudo-Riemannian manifold of dimension $n\ge 2$. A projective transformation is a diffeomorphism of $M^n$ that maps unparameterized geodesics to geodesics. Let…

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…

Nonexistence conjecture. No such isometric immersion is possible for any $\tilde{c}>-1$.

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 $V$ be a finite-dimensional real vector space, let $\langle\cdot,\cdot\rangle$ be a non-degenerate symmetric bilinear form on $V$, and let $A$ be an algebraic curvature tensor.…

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…

Conjecture on compact negatively curved space forms. Such a compact manifold exists if and only if $(p,q)$ occurs in the list

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

Let $\mathcal{M}:=(M,g)$ be a pseudo-Riemannian manifold of Lorentzian signature $(1,q)$. For $P\in M$, let $g_P:=g|_{T_PM}$ and let $\nabla^iR_P:=\nabla^iR|_{T_PM}$. Its $k$-model…

The paper considers functions on pseudo-Riemannian symmetric spaces and a proposed series of non-$L^2$ inner products on those spaces. Neretin's conjecture. These inner products sh…

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

The esc conjecture. Every connected component of the space of Lorentzian metrics on $M$ contains a metric with scalar curvature $s$.

Consider two locally controllable gradient systems, with state spaces equipped with their underlying pseudo-Riemannian manifolds, and let a state space diffeomorphism identify thei…

Colin de Verdière–Le Bihan conjecture. The operator $D$ is essentially self-adjoint if and only if the Hamiltonian flow of its symbol is complete.

Inequivalent indefinite metrics conjecture. For a given Lie algebra $\mathfrak g$, there are a number of inequivalent indefinite metrics on $\mathfrak g$.

A compact pseudo-Riemannian conformal manifold is essential when its conformal group is not reducible to an isometric action for any metric in the conformal class. It is conformall…

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…

D’Ambra–Gromov conjecture. Compact pseudo-Riemannian conformal manifolds with an essential action of the conformal group should be conformally flat.

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…

The spacetime manifold is endowed with a projective structure by the principle of universality of free fall, meaning an equivalence class of curves up to reparametrization, and wit…