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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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Good books on Geometric Theory of Dynamical Systems

Mainly from the Hamiltonian point of view: Zehnder: Lectures on Dynamical Systems
4 votes

Is there a sheaf theoretical characterization of a differentiable manifold?

This is more a long comment than an answer, but the comments concerning the question Hausdorff or non-Hausdorff triggered this... The second countable condition is certainly desirable for many reason …
Stefan Waldmann's user avatar
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On the topology induced by a Lorentzian metric

In general, this topology is coarser than the original topology of the manifold, and, without further assumptions, strictly coarser. It coincides with the original one iff the Lorentz manifold is stro …
Stefan Waldmann's user avatar