The manifold of orthonormal -frames in an -dimensional Euclidean space. In a similar way one defines a complex Stiefel manifold and a quaternion Stiefel manifold . Stiefel manifolds are compact real-analytic manifolds, and also homogeneous spaces of the classical compact groups , and , respectively. In particular, , , are the spheres, the Stiefel manifold is the manifold of unit vectors tangent to , the Stiefel manifolds , , are identified with the groups , , , and — with the group . Sometimes non-compact Stiefel manifolds, consisting of all possible -frames in , or , are considered.
These manifolds were introduced by E. Stiefel  in connection with systems of linearly independent vector fields on smooth manifolds. First started in , studies on the topology of Stiefel manifolds led later to the complete calculation of their cohomology rings (see , ). In particular,
is a commutative algebra with generators and relations
(everywhere above, denotes an element of order ). Real, complex and quaternion Stiefel manifolds are aspherical in dimensions not exceeding , and , respectively. Moreover,
The computation of other homotopy groups of Stiefel manifolds is discussed in .
|||E. Stiefel, "Richtungsfelder und Fernparallelismus in -dimensionalen Mannigfaltigkeiten" Comm. Math. Helv. , 8 : 4 (1935–1936) pp. 305–353|
|||A. Borel, , Fibre spaces and their applications , Moscow (1958) pp. 163–246 (In Russian; translated from French)|
|||N.E. Steenrod, D.B.A. Epstein, "Cohomology operations" , Princeton Univ. Press (1962)|
|||V.A. Rokhlin, D.B. Fuks, "Beginner's course in topology. Geometric chapters" , Springer (1984) (Translated from Russian)|
|||Itogi Nauk. Algebra. Topol. Geom. (1971) pp. 71–122|
For homotopy groups of Stiefel manifolds see also .
Another (and better) frequently used notation for the Stiefel manifolds , and is , , , generalizing to where is an appropriate vector space.
As homogeneous spaces these Stiefel manifolds are equal to, respectively,
The natural quotient mapping , etc., assigns to an orthogonal, etc., matrix the -frame consisting of its first columns.
There are canonical mappings from the Stiefel manifolds to the Grassmann manifolds (cf. Grassmann manifold):
which assign to a -frame the -dimensional subspace spanned by that frame. This exhibits the Grassmann manifolds as homogeneous spaces:
Given an -dimensional (real, complex, quaternionic) vector bundle over a space , the associated Stiefel bundles have the fibres over , where is the fibre of over . Similarly one has the Grassmann bundle , whose fibre over is the Grassmann manifold .
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Stiefel manifold. A.L. Onishchik (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Stiefel_manifold&oldid=12028