7 problems

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…

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

Let $k\geq 3$. Let $E=S^n\times\mathbb{R}^k$ carry a complete metric with nonnegative sectional curvature, and let $\Sigma$ be its soul. Consider the boundary of a small metric tub…

Finsler soul conjecture. $M$ is diffeomorphic to

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…

Soul neighborhood conjecture. The space $X$ is homeomorphic to a small neighborhood of $S$ that has the structure of an open disk bundle over $S$.

Soul bundle conjecture. There exists a positive number $\epsilon$ such that