In mathematics, the Stiefel manifold Vk(Rn) is the set of all orthonormal k-frames in Rn. That is, it is the set of ordered k-tuples of orthonormal vectors in Rn. It is named after Swiss mathematician Eduard Stiefel. Likewise one can define the complex Stiefel manifold Vk(Cn) of orthonormal k-frames in Cn and the quaternionic Stiefel manifold Vk(Hn) of orthonormal k-frames in Hn. More generally, the construction applies to any real, complex, or quaternionic inner product space.
In some contexts, a non-compact Stiefel manifold is defined as the set of all linearly independent k-frames in Rn, Cn, or Hn; this is homotopy equivalent, as the compact Stiefel manifold is a deformation retract of the non-compact one, by Gram–Schmidt. Statements about the non-compact form correspond to those for the compact form, replacing the orthogonal group (or unitary or symplectic group) with the general linear group.
Read more about Stiefel Manifold: Topology, As A Homogeneous Space, Special Cases, As A Principal Bundle, Homotopy
Famous quotes containing the word manifold:
“As one who knows many things, the humanist loves the world precisely because of its manifold nature and the opposing forces in it do not frighten him. Nothing is further from him than the desire to resolve such conflicts ... and this is precisely the mark of the humanist spirit: not to evaluate contrasts as hostility but to seek human unity, that superior unity, for all that appears irreconcilable.”
—Stefan Zweig (18811942)