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,
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is a commutative algebra with generators
and relations
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(everywhere above, denotes an element of order
). Real, complex and quaternion Stiefel manifolds are aspherical in dimensions not exceeding
,
and
, respectively. Moreover,
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The computation of other homotopy groups of Stiefel manifolds is discussed in [5].
References
[1] | E. Stiefel, "Richtungsfelder und Fernparallelismus in ![]() |
[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,
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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):
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which assign to a -frame the
-dimensional subspace spanned by that frame. This exhibits the Grassmann manifolds as homogeneous spaces:
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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 ![]() |
[a3b] | G.F. Paechter, "The groups ![]() |
[a3c] | G.F. Paechter, "The groups ![]() |
[a3d] | G.F. Paechter, "The groups ![]() |
[a4] | M.W. Hirsch, "Differential topology" , Springer (1976) pp. 4, 78 |
[a5] | J.W. Milnor, J.D. Stasheff, "Characteristic classes" , Princeton Univ. Press (1974) |
Stiefel manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stiefel_manifold&oldid=12028