12 problems

Wilhelm's conjecture. Every such submersion satisfies

Let $\pi\colon (M^{n+k},\textsl g)\to (B^n,b)$ be a Riemannian submersion. Suppose that $(M,\textsl g)$ is compact and has positive sectional curvature. Petersen–Wilhelm's conjectu…

Pu–Loewner equality rigidity conjecture. Both the topology and the metric must be extremal: (a) $X$ fibers smoothly over $\mathbb T^2$ with fiber $\mathbb R P^2$; (b)…

Pu–Loewner inequality conjecture. Every such metric satisfies

Weak-integrability conjectures. (1) If $\partial Y=\emptyset$ and $f$ is weakly integrable, then $f$ is integrable. (2) If $\Sigma\in \operatorname{Alex}^n(1)$,…

Let $F\hookrightarrow M\rightarrow B$ be a Riemannian submersion from a positively curved closed manifold $M$. Petersen–Wilhelm's fiber dimension conjecture. Then … This conjecture…

Petersen–Wilhelm conjecture. The inequalities

Grove's conjecture. A Riemannian submersion

Let $M^m(c)$ be an $m$-dimensional space form of constant sectional curvature $c$, let $N^{m-1}$ be a manifold, and let $\varphi:M^m(c)\to N^{m-1}$ be a Riemannian submersion. Subm…

Let $(M^4,g)$ be a compact four-dimensional Riemannian manifold with positive sectional curvature. A nontrivial Riemannian submersion is a Riemannian submersion from $M^4$ whose ba…

Let $M$ be a $(2n+1)$-dimensional semi-Riemannian space form $M_{1}^{2n+1}(c)$ of index $1$, and let $B$ be a $2n$-dimensional Riemannian manifold. A proper biharmonic semi-Riemann…

Let $G$ be a simple compact Lie group with a bi-invariant metric, and let $\pi:G\rightarrow B$ be a Riemannian submersion with connected totally geodesic fibers. Coset-fibration co…