A spin structure on a smooth orientable Riemannian manifold is a lift of the principal O(n) bundle associated to its tangent bundle to a principal Spin(n) bundle. Such a lift exists if and only if the second Stiefel-Whitney class of the tangent bundle vanishes. Smooth could mean many things, including: Smooth function, a function that is infinitely differentiable, used in calculus and topology. ... This article or section should be merged with Orientable manifold. ... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... In mathematics, a principal G-bundle is a special kind of fiber bundle for which the fibers are all G-torsors (also known as principal homogeneous spaces) for the action of a topological group G. Principal G-bundles are G-bundles in the sense that the group G also serves... In mathematics, the tangent bundle of a differentiable manifold is a vector bundle which as a set is the disjoint union of all the tangent spaces at every point in the manifold with natural topology and smooth structure. ... In mathematics, a principal G-bundle is a special kind of fiber bundle for which the fibers are all G-torsors (also known as principal homogeneous spaces) for the action of a topological group G. Principal G-bundles are G-bundles in the sense that the group G also serves...

See also Spin^c structure.