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In mathematics, spin geometry is the area of differential geometry and topology where objects like spin manifolds and Dirac operators, and the various associated index theorems have come to play a fundamental role both in mathematics and in mathematical physics.
In differential geometry, a spin structure on an orientable Riemannian manifold (M, g) allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry. Spin structures have wide applications to mathematical physics , in particular to quantum field theory where they are an essential ingredient in ...
Spin is an intrinsic form of angular momentum carried by elementary particles, and thus by composite particles such as hadrons, atomic nuclei, and atoms. [1] [2]: 183–184 Spin is quantized, and accurate models for the interaction with spin require relativistic quantum mechanics or quantum field theory.
Spinors were introduced in geometry by Élie Cartan in 1913. [1] [d] In the 1920s physicists discovered that spinors are essential to describe the intrinsic angular momentum, or "spin", of the electron and other subatomic particles. [e] Spinors are characterized by the specific way in which they behave under rotations.
In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition. [ 1 ] By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance (so it is an isometry ), and orientation ...
In mathematics the spin group, denoted Spin(n), [1] [2] is a Lie group whose underlying manifold is the double cover of the special orthogonal group SO(n) = SO(n, R), such that there exists a short exact sequence of Lie groups (when n ≠ 2)
The Seiberg–Witten invariant of a four-manifold M with b 2 + (M) ≥ 2 is a map from the spin c structures on M to Z. The value of the invariant on a spin c structure is easiest to define when the moduli space is zero-dimensional (for a generic metric). In this case the value is the number of elements of the moduli space counted with signs.
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space , the n {\displaystyle n} -sphere , hyperbolic space , and smooth surfaces in three-dimensional space, such as ellipsoids and paraboloids , are all examples of ...