17 problems

A foliated manifold $(M,\mathcal{F})$ is taut if $M$ admits a Riemannian metric for which every leaf of $\mathcal{F}$ is a minimal submanifold. The foliation is transversely orient…

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…

Let $\mathcal{F}$ be a $q$-codimensional complete Riemannian foliation of a connected manifold $M$, with transverse sectional curvature satisfying $1/4<\sec_{\mathcal{F}}\leq 1$ an…

Wilking's conjecture. Every transnormal system is a singular Riemannian foliation.

Orbifold representation conjecture. If such an orbifold exists, then it must be a very good, orientable orbifold with isolated singularities.

Transnormal-system smoothness conjecture. Every transnormal system is a smooth foliation.

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 $\mathcal{F}$ be a complete singular Riemannian foliation, and let $\mathscr{C}_{\mathcal{F}}$ be the locally constant sheaf of Lie algebras associated with its structural theo…

Carrière's conjecture. A Riemannian foliation on a compact manifold is unimodular if and only if it is taut.

Let $(M,g)$ be a Riemannian manifold endowed with a Riemannian foliation, and let conclusion (2) refer to the preceding assertion that, under the stated adiabatic collapse hypothes…

Grove's conjecture. A Riemannian submersion

Let $\cal F$ be a Riemannian foliation with bounded holonomy, and suppose that no leaf has infinite fundamental group. Let $H_p$ denote the holonomy group at $p$. Bounded-holonomy…

Let $\cal F$ be a Riemannian foliation, let $H_p$ denote the holonomy group at $p$, and let $\mathcal V_p$ be the corresponding vertical space. The inclusion…

Let $M$ be a compact manifold with positive curvature, and let $\cal F$ be a Riemannian foliation on $M$. A horizontal vector is a vector tangent to the horizontal distribution of…

A foliated manifold is uniform if its leaf space is compact. A Riemannian foliation is complete when it admits a complete bundle-like Riemannian metric, and it is strongly tense wh…

Nonexistence conjecture. There are no $7$-dimensional regular Riemannian foliations on $S^{15}$ other than the Hopf fibration

Let $(M,\mathcal{F})$ be a Riemannian foliation, let $L$ be the generalized basic Laplacian, and suppose the heat operators $e^{-tL^{+}}$ and $e^{-tL^{-}}$ have asymptotic expansio…