Stiefel manifold

(real)

The manifold $ V _ {n,k} $ of orthonormal $ k $- frames in an $ n $- dimensional Euclidean space. In a similar way one defines a complex Stiefel manifold $ W _ {n,k} $ and a quaternion Stiefel manifold $ X _ {n,k} $. Stiefel manifolds are compact real-analytic manifolds, and also homogeneous spaces of the classical compact groups $ O( n) $, $ U( n) $ and $ \mathop{\rm Sp} ( n) $, respectively. In particular, $ V _ {n,1} = S ^ {n-} 1 $, $ W _ {n,1} = S ^ {2n-} 1 $, $ X _ {n,1} = S ^ {4n-} 1 $ are the spheres, the Stiefel manifold $ V _ {n,2} $ is the manifold of unit vectors tangent to $ S ^ {n-} 1 $, the Stiefel manifolds $ V _ {n,n} $, $ W _ {n,n} $, $ X _ {n,n} $ are identified with the groups $ O( n) $, $ U( n) $, $ \mathop{\rm Sp} ( n) $, and $ V _ {n,n-} 1 $— with the group $ \mathop{\rm SO} ( n) $. Sometimes non-compact Stiefel manifolds, consisting of all possible $ k $- frames in $ \mathbf R ^ {n} $, $ \mathbf C ^ {n} $ or $ \mathbf H ^ {n} $, 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,

$$ H ^ \star ( W _ {n,k} , \mathbf Z ) = \Lambda _ {\mathbf Z} ( x _ {2n-} 1 , x _ {2n-} 3 \dots x _ {2(} n- k)+ 1 ), $$

$$ H ^ \star ( X _ {n,k} , \mathbf Z ) = \Lambda _ {\mathbf Z} ( x _ {4n-} 1 , x _ {4n-} 5 \dots x _ {4(} n- k)+ 3 ), $$

$ H ^ \star ( V _ {n,k} , \mathbf Z _ {2} ) $ is a commutative algebra with generators $ x _ {n-} k \dots x _ {n-} 1 $ and relations

$$ x _ {i} x _ {j} = \left \{ \begin{array}{lll} x _ {i+} j & \textrm{ for } &i+ j \leq n- 1, \\ 0 & \textrm{ for } &i+ j > n- 1 \\ \end{array} \right .$$

(everywhere above, $ x _ {l} $ denotes an element of order $ l $). Real, complex and quaternion Stiefel manifolds are aspherical in dimensions not exceeding $ n- k- 1 $, $ 2( n- k) $ and $ 4( n- k)+ 2 $, respectively. Moreover,

$$ \pi _ {n-} k ( V _ {n,k} ) \cong \left \{ \begin{array}{ll} \mathbf Z & \textrm{ if } k= 1 \textrm{ or } n- k \textrm{ is even }, \\ \mathbf Z _ {2} & \textrm{ if } k> 1 \textrm{ or } n- k \textrm{ is odd } ; \\ \end{array} \right .$$

$$ \pi _ {2(} n- k)+ 1 ( W _ {n,k} ) \cong \pi _ {4(} n- k)+ 3 ( X _ {n,k} ) \cong \mathbf Z . $$

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 $ V _ {n,k } $, $ W _ {n,k } $ and $ X _ {n,k } $ is $ V _ {k} ( \mathbf R ^ {n} ) $, $ V _ {k} ( \mathbf C ^ {n} ) $, $ V _ {k} ( \mathbf H ^ {n} ) $, generalizing to $ V _ {k} ( E ) $ where $ E $ is an appropriate vector space.

As homogeneous spaces these Stiefel manifolds are equal to, respectively,

$$ V_{k} ( \mathbf R^n) = \frac{O(n)}{O(n- k)} = \frac{ \mathop{\rm SO} (n)}{\mathop{\rm SO} (n-k)}, $$

$$ V_{k} ( \mathbf C ^{n}) = \frac{U(n)}{U(n- k)} = \frac{ \mathop{\rm SU} (n)}{ \mathop{\rm SU} (n-k)}, $$

$$ V _ {k} ( \mathbf H ^ {n} ) = \frac{ \mathop{\rm Sp} ( n) }{ \mathop{\rm Sp} ( n- k) }. $$

The natural quotient mapping $ O( n) \rightarrow V _ {k} ( \mathbf R ^ {n} ) $, etc., assigns to an orthogonal, etc., matrix the $ k $- frame consisting of its first $ k $ columns.

There are canonical mappings from the Stiefel manifolds to the Grassmann manifolds (cf. Grassmann manifold):

$$ V _ {k} ( E) \rightarrow \mathop{\rm Gr}_{k} ( E) , $$

which assign to a $ k $- frame the $ k $- dimensional subspace spanned by that frame. This exhibits the Grassmann manifolds as homogeneous spaces:

$$ \mathop{\rm Gr}_{k} (\mathbf R^{n} ) = \frac {O(n)}{O(k)\times O(n-k)} , $$

etc.

Given an $ n $- dimensional (real, complex, quaternionic) vector bundle $ E $ over a space $ X $, the associated Stiefel bundles $ V _ {k} ( E) $ have the fibres $ V _ {k} ( E _ {x} ) $ over $ x \in X $, where $ E _ {x} $ is the fibre of $ E $ over $ x $. Similarly one has the Grassmann bundle $ \mathop{\rm Gr} _ {k} ( E) $, whose fibre over $ x \in X $ is the Grassmann manifold $ \mathop{\rm Gr} _ {k} ( E _ {x} ) $.

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 $\pi_r(V_{n,m})$" Quarterly J. Math. , 7 (1956) pp. 249–268
[a3b]	G.F. Paechter, "The groups $\pi_r(V_{n,m})$" Quarterly J. Math. , 9 (1958) pp. 8–27
[a3c]	G.F. Paechter, "The groups $\pi_r(V_{n,m})$" Quarterly J. Math. , 10 (1959) pp. 17–37; 241–260
[a3d]	G.F. Paechter, "The groups $\pi_r(V_{n,m})$" 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)