51 problems

A rationally elliptic space is a simply connected topological space $X$ such that … A smooth manifold is said to have nonnegative sectional curvature when it admits a Riemannian me…

Let $M^n$ be a manifold with nonnegative sectional curvature. A group is virtually abelian if it has an abelian subgroup of finite index. Fukaya–Yamaguchi conjecture. The fundament…

Grove–Ziller conjecture. Any cohomogeneity one manifold admits a Riemannian metric with non-negative sectional curvature.

A double disk bundle decomposition writes a smooth manifold as the union of two disk bundles glued along their common boundary by a diffeomorphism. The conjecture concerns closed,…

Let $T^k$ act isometrically and effectively on $M^n$, where $M^n$ is a closed, simply connected, non-negatively curved Riemannian manifold. Define … where $n_i\geq 2$,…

Let $M$ be a closed simply connected manifold admitting a Riemannian metric of nonnegative sectional curvature. Grove's conjecture. Then $M$ has a DDBD structure, that is, an isopa…

Maximal Symmetry Rank Conjecture.

Intrinsic characterization conjecture. $(\mathbb S^n,g)$ admits a smooth global isometric embedding

Let $G$ be a Lie group with bi-invariant metrics $h_0$ and $h_1$, let $M$ be the matrix of $h_1$ with respect to $h_0$, and let $h$ be another invariant metric. Write $\Phi$ for th…

The orthogonal abelian-subalgebra conjecture. One has $\lambda=\mu=0$. Equivalently, $\mathfrak{g}$ decomposes into three orthogonal $\Psi$-invariant abelian subalgebras. The theor…

Let $G=SO(4)$ with Lie algebra $\mathfrak{g}=so(4)=\mathfrak{g}_1\oplus\mathfrak{g}_2$, where $\mathfrak{g}_1$ and $\mathfrak{g}_2$ are the two factors. Let…

Assume that $M$ is a compact simply connected 5-manifold with a metric of non-negative sectional curvature. The known models are $\mathbb{S}^5$,…

Invariant-abelian-subalgebra conjecture. The parameters satisfy $\lambda=\mu=0$. Equivalently, $\mathfrak{g}$ decomposes into three orthogonal $\Psi$-invariant abelian subalgebras.

S^3-action rigidity conjecture. If $\Psi$ has no nonzero singular eigenvectors, then $\Psi$ is a known example coming from an $S^3$-action.

Inverse-linear conjecture. If $h$ has nonnegative curvature, then the inverse-linear path from $h_0$ to $h_1=h$ consists entirely of metrics with nonnegative curvature.

Let $G$ be a compact Lie group with a bi-invariant metric $h_0$ and Lie algebra $\mathfrak g$. For a left-invariant metric $h$ on $G$, let $\Phi$ be the unique $h_0$-self-adjoint p…

Let $M$ be a complete manifold with nonnegative sectional curvature, and let $\mathcal{F}$ be a singular Riemannian foliation on $M$. The dual-foliation completeness conjecture. Ev…

Félix–Halperin–Thomas ellipticity conjecture. Manifolds of nonnegative sectional curvature are elliptic.

Let $S^n\times\mathbb{R}^k$ carry a complete metric of nonnegative sectional curvature, with soul and a small metric tube about the soul. Weak Hopf Conjecture. There does not exist…

Normal holonomy conjecture. The normal holonomy group is isomorphic to a subgroup of $\operatorname{SO}(n)$.

A simply connected closed Riemannian manifold is integrally elliptic if its integral homotopy is constrained in the sense intended by the integral ellipticity conjecture. Integral…

Gromov's conjecture. The total Betti number should satisfy

A good complexification of a closed smooth manifold $M$ is a smooth affine algebraic variety $U$ over $\mathbb{R}$ such that … and the inclusion … is a homotopy equivalence. Good c…

Splitting conjecture. If $\phi$ is integrable, then

Soul conjecture. If $X$ contains an open subset where the curvature is bounded below by a positive constant, then $S$ is a point.