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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”.
9
votes
Accepted
Lorentzian metrics on the torus up to continuos deformations
Yes those are already all different metrics. Since the tangent bundle of the 2-torus $\mathbb{T}^2$ is trivial you have a correspondence between the set of homotopy classes of maps $\mathbb{T}^2\to\ma …
2
votes
Spin structures and divisibility of cohomology classes
I think your last sentence/question is not well posed. It is well known that there are flat manifolds admitting plenty of non-flat vector bundles. E.g. see J. Smillie, "Flat manifolds with non-zero Eu …