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The study of differentiable manifolds and differentiable maps. One fundamental problem is that of classifying manifolds up to diffeomorphism. Differential topology is what Poincaré understood as topology or “analysis situs”.
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Locally conformally flat
The connected sum of two locally conformally flat manifolds can be equipped with a metrics which is locally conformally flat. To see this, recall that the product metric on $\mathbb{R} \times S^{n-1} …