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

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Stiefel,   "Richtungsfelder und Fernparallelismus in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777038.png" />-dimensionalen Mannigfaltigkeiten"  ''Comm. Math. Helv.'' , '''8''' :  4  (1935–1936)  pp. 305–353</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Borel,  , ''Fibre spaces and their applications'' , Moscow  (1958)  pp. 163–246  (In Russian; translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.E. Steenrod,   D.B.A. Epstein,   "Cohomology operations" , Princeton Univ. Press  (1962)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.A. Rokhlin,   D.B. Fuks,   "Beginner's course in topology. Geometric chapters" , Springer  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  ''Itogi Nauk. Algebra. Topol. Geom.''  (1971)  pp. 71–122</TD></TR></table>

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