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I was interested in whether a manifold which admits a metric of constant sectional curvature can be homotopy equivalent to a product of non-contractible manifolds. Of course, there are three cases: ...

Put $G= \mathbb{C} \rtimes_{\phi} \mathbb{C} \setminus \{0\}$ where $\phi_{a} (z)= az$ for $a \in \mathbb{C} \setminus \{0\}$. $G$ is a real $4$ dimensional Lie group; then it has a unique left ...