Stiefel manifold

(real)

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 [1] in connection with systems of linearly independent vector fields on smooth manifolds. First started in [1], studies on the topology of Stiefel manifolds led later to the complete calculation of their cohomology rings (see [2], [3]). 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 [5].

References

[1]	E. Stiefel, "Richtungsfelder und Fernparallelismus in -dimensionalen Mannigfaltigkeiten" Comm. Math. Helv. , 8 : 4 (1935–1936) pp. 305–353
[2]	A. Borel, , Fibre spaces and their applications , Moscow (1958) pp. 163–246 (In Russian; translated from French)
[3]	N.E. Steenrod, D.B.A. Epstein, "Cohomology operations" , Princeton Univ. Press (1962)
[4]	V.A. Rokhlin, D.B. Fuks, "Beginner's course in topology. Geometric chapters" , Springer (1984) (Translated from Russian)
[5]	Itogi Nauk. Algebra. Topol. Geom. (1971) pp. 71–122

Comments

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:

etc.

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 .

References

[a1]	D. Husemoller, "Fibre bundles" , McGraw-Hill (1966)
[a2]	J. Dieudonné, "A history of algebraic and differential topology 1900–1960" , Birkhäuser (1989)
[a3a]	G.F. Paechter, "The groups " Quarterly J. Math. , 7 (1956) pp. 249–268
[a3b]	G.F. Paechter, "The groups " Quarterly J. Math. , 9 (1958) pp. 8–27
[a3c]	G.F. Paechter, "The groups " Quarterly J. Math. , 10 (1959) pp. 17–37; 241–260
[a3d]	G.F. Paechter, "The groups " Quarterly J. Math. , 11 (1960) pp. 1–16
[a4]	M.W. Hirsch, "Differential topology" , Springer (1976) pp. 4, 78
[a5]	J.W. Milnor, J.D. Stasheff, "Characteristic classes" , Princeton Univ. Press (1974)