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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”.

3 votes

Differentiable structures on R^3

There are infinitely many differentiable structures on $\mathbb R$ : take any homeomorphism which is no diffeomorphism (such as $x\mapsto x^3$), and you get an non-usual differentiable structure on $\ …
hrkrshnn's user avatar
  • 239
4 votes

What's the difference between a real manifold and a smooth variety?

I think there's some big difference concerning the metric approach too. In fact, the Gram-Schmidt process (which is real analytic) enables us -in real differential geometry- to find some local orthon …
Henri's user avatar
  • 2,727