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
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