23 problems

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

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 $S^n\times\mathbb{R}^k$ 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 $\operatorname{SO}(n)$.

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.

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 smooth manifold admitting a Riemannian metric of non-negative sectional curvature. A double disk bundle is a smooth closed manifold obtained by…

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

Let $M$ be a $2$-connected $7$-manifold whose cohomology ring is that of an $\mathbf{S}^3$-bundle over $\mathbf{S}^4$. A non-negative curvature and Seifert fibration conjecture ass…

Alternation conjecture. The odd GKM$_3$ graph corresponding to $M^{2n+1}$ is alternating.

Let $M$ be an isoparametric hypersurface or a focal submanifold of the unit sphere $\mathbb{S}^n$. Tang's conjecture. Every such isoparametric hypersurface and focal submanifold ad…

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. One has

Standard-manifold classification conjecture. The manifold $N$ is diffeomorphic to one of the $5$ standard manifolds

Finsler soul conjecture. $M$ is diffeomorphic to

Let a compact Lie group act smoothly and isometrically on a simply connected compact manifold of nonnegative curvature. The action is irreducible when its associated reflection gro…

Let $M$ be a closed simply connected Riemannian manifold with non-negative sectional curvature. A singular Riemannian foliation of codimension 1 is a singular Riemannian foliation…

Let $A\in \operatorname{Alex}^n(0)$ be a noncompact Alexandrov space with a soul $S$. Let $C_i$ be the subsets associated with the soul as in 1.1, and let…

Let $A$ and $A'$ be nonnegatively curved Alexandrov spaces with isometric boundaries, and let $A\cup_\partial A'$ denote their gluing along a boundary isometry. Let $D^3$,…

A polar action is an isometric group action admitting a section, a submanifold meeting every orbit orthogonally. Let a compact simply connected manifold have nonnegative sectional…

Let $M$ be an orientable closed simply connected nonnegatively curved $4$-manifold with an effective isometric $S^1$-action. Hsiang–Kleiner's conjecture. $M$ is diffeomorphic to ei…