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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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A possible characterization of sphere or projective space
The article by X. Liu and Sh. Deng "The antipodal sets of compact symmetric spaces" gives many examples, e.g. $\mathrm{SU}(2n)$, $\mathrm{Spin}(5)$, $\mathrm{Spin}(7)$,.... All those spaces have uniqu …