principal bundle
fiber bundle whose fibers are group torsors (groups with the identity element forgotten)
spin structure
lift of the SO(n) frame bundle of an oriented Riemannian manifold into Spin(n), the spin group
Stiefel manifold
the manifold of all orthonormal k-frames in n-dimensional Euclidean space
frame bundle
principal bundle associated to a vector bundle, whose fiber is the (torsor over the) automorphism group of the vector-bundle fiber