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I know classification of 2 manifolds and geometrization for 3 manifolds. Why for dimension great or equal to 4, this task become impossible? edit: Or should I ask "why geometrization won't be possibl …

For an open complete Riemannian manifold $M$ with non-negative sectional curvature, the Busemann function defined below is a convex exhaustion function (by Cheeger-Gromoll's proof of soul theorem) Th …

Let $M$ be a open complete manifold with Ricci curvature $\ge 0$. By a theorem of Calabi and Yau, the volume growth of $M$ is at least of linear. I am wondering whether the following statement is true …

Let $M$ be a closed $n$-dimensional Riemannian manifold with positive sectional curvature. Let $G$ be a close subgroup of isometry group ${\rm Iso}(M)$. Suppose the action of $G$ on $M$ is not transit …

Let $E$ be the total space of the sphere bundle $S^k\to E\to M$, is it true that there exists a disk bundle $D^{k+1}\to N\to M$ such that $E=\partial N$? (where $D^{k+1}$ is the unit disk in $\mathbb …