spin geometry
An electron, turned all the way around by 360 degrees, comes back not to itself but to its negative; you have to turn it 720 degrees to get back to start. That strange double-valued behaviour is 'spin', and it is not a quirk of physics but a fact about the topology of rotation groups. Spin geometry is the part of differential geometry that takes this seriously: it studies manifolds equipped with a consistent global way of talking about spinors, and the operators (above all the Dirac operator) that act on them. It is where geometry, the topology of vector bundles, and quantum physics meet most tightly.
The technical heart is the spin group and the spin structure. The rotation group SO(n) is not simply connected — its fundamental group is Z/2 for n >= 3 — so it has a connected double cover, the spin group Spin(n), and representations of Spin(n) that do not descend to SO(n) are spinor representations. A spin structure on an oriented Riemannian manifold M is a lift of its orthonormal frame bundle (a principal SO(n)-bundle) to a principal Spin(n)-bundle, compatible with the double cover. Such a lift exists if and only if the second Stiefel-Whitney class w_2(M) in H^2(M; Z/2) vanishes, and when it exists the inequivalent choices form a torsor over H^1(M; Z/2). Given a spin structure you build the spinor bundle, Clifford multiplication, and the Dirac operator; the algebra of all this lives in the Clifford algebra Cl(n), whose representation theory (and its mod-8 periodicity) governs how spinors behave in each dimension. A weaker but always-available notion on oriented 4-manifolds is a spin-c structure, which is what Seiberg-Witten theory uses.
Why it matters: spin geometry is the natural home of the Atiyah-Singer index theorem in its sharpest form, of positive-scalar-curvature obstructions via the Lichnerowicz formula, and of the entire Seiberg-Witten revolution in four-manifold topology. The honest caveats are topological. First, not every manifold is spin — the real projective plane and many others are not — so spin geometry is conditional, and you must check w_2 before invoking a Dirac operator. Second, 'spin' is extra structure, not a property read off the metric alone: a manifold can admit several inequivalent spin structures (the circle has two), and they genuinely give different Dirac operators and different spectra. Do not conflate orientable, spin, and spin-c; they are nested but distinct conditions (spin implies spin-c implies orientable, and none of the reverse implications hold).
The circle S^1 has SO(1) trivial but its frame bundle still admits two inequivalent spin structures, the 'bounding' (antiperiodic) and 'nonbounding' (periodic) ones, indexed by H^1(S^1; Z/2) = Z/2; the resulting Dirac operators d/dt have different spectra (half-integers versus integers), the simplest example of inequivalent spin structures giving different geometry.
Two spin structures on the circle yield Dirac spectra of half-integers versus integers — spin structure is genuine extra data.
A frequent slip is to assume every oriented manifold is spin; the obstruction is w_2(M), and many natural spaces (e.g. CP^2) are not spin though they are spin-c, which is why four-manifold gauge theory works with spin-c rather than spin structures.