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''(real)''
 
''(real)''
  
The manifold $  V _ {n,k} $
+
The manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s0877701.png" /> of orthonormal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s0877702.png" />-frames in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s0877703.png" />-dimensional Euclidean space. In a similar way one defines a complex Stiefel manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s0877704.png" /> and a quaternion Stiefel manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s0877705.png" />. Stiefel manifolds are compact real-analytic manifolds, and also homogeneous spaces of the classical compact groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s0877706.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s0877707.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s0877708.png" />, respectively. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s0877709.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777011.png" /> are the spheres, the Stiefel manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777012.png" /> is the manifold of unit vectors tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777013.png" />, the Stiefel manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777016.png" /> are identified with the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777019.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777020.png" /> — with the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777021.png" />. Sometimes non-compact Stiefel manifolds, consisting of all possible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777022.png" />-frames in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777024.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777025.png" />, are considered.
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,
 
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,
  
$$
+
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H  ^  \star  ( W _ {n,k} , \mathbf Z )  = \Lambda _ {\mathbf Z} ( x _ {2n-} 1 , x _ {2n-} 3 \dots x _ {2(} n- k)+ 1 ),
 
$$
 
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777027.png" /></td> </tr></table>
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} ) $
+
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777028.png" /> is a commutative algebra with generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777029.png" /> and relations
is a commutative algebra with generators $  x _ {n-} k \dots x _ {n-} 1 $
 
and relations
 
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777030.png" /></td> </tr></table>
x _ {i} x _ {j}  = \left \{
 
  
(everywhere above, $  x _ {l} $
+
(everywhere above, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777031.png" /> denotes an element of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777032.png" />). Real, complex and quaternion Stiefel manifolds are aspherical in dimensions not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777035.png" />, respectively. Moreover,
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,
 
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777036.png" /></td> </tr></table>
\pi _ {n-} k ( V _ {n,k} )  \cong  \left \{
 
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777037.png" /></td> </tr></table>
\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]]].
 
The computation of other homotopy groups of Stiefel manifolds is discussed in [[#References|[5]]].
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====References====
 
====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>
 
<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====
 
====Comments====
 
For homotopy groups of Stiefel manifolds see also .
 
For homotopy groups of Stiefel manifolds see also .
  
Another (and better) frequently used notation for the Stiefel manifolds $  V _ {n,k }  $,
+
Another (and better) frequently used notation for the Stiefel manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777041.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777044.png" />, generalizing to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777045.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777046.png" /> is an appropriate vector space.
$  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,
 
As homogeneous spaces these Stiefel manifolds are equal to, respectively,
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777047.png" /></td> </tr></table>
V _ {k} ( \mathbf R  ^ {n} )  = \
 
O(
 
\frac{n)}{O(}
 
n- k)  = \
 
 
 
\frac{ \mathop{\rm SO} ( n) }{ \mathop{\rm SO} ( n- k) }
 
,
 
$$
 
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777048.png" /></td> </tr></table>
V _ {k} ( \mathbf C  ^ {n} )  = U(
 
\frac{n)}{U(}
 
n- k)  =
 
\frac{ \mathop{\rm SU} ( n) }{ \mathop{\rm SU} ( n- k) }
 
,
 
$$
 
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777049.png" /></td> </tr></table>
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} ) $,  
+
The natural quotient mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777050.png" />, etc., assigns to an orthogonal, etc., matrix the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777051.png" />-frame consisting of its first <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777052.png" /> columns.
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|Grassmann manifold]]):
 
There are canonical mappings from the Stiefel manifolds to the Grassmann manifolds (cf. [[Grassmann manifold|Grassmann manifold]]):
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777053.png" /></td> </tr></table>
V _ {k} ( E)  \rightarrow  \mathop{\rm Gr} _ {k} ( E) ,
 
$$
 
  
which assign to a $  k $-
+
which assign to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777054.png" />-frame the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777055.png" />-dimensional subspace spanned by that frame. This exhibits the Grassmann manifolds as homogeneous spaces:
frame the $  k $-
 
dimensional subspace spanned by that frame. This exhibits the Grassmann manifolds as homogeneous spaces:
 
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777056.png" /></td> </tr></table>
\mathop{\rm Gr} _ {k} ( \mathbf R  ^ {n} )  = \
 
O(
 
\frac{n)}{O(}
 
k)\times O( n- k) ,
 
$$
 
  
 
etc.
 
etc.
  
Given an $  n $-
+
Given an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777057.png" />-dimensional (real, complex, quaternionic) [[Vector bundle|vector bundle]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777058.png" /> over a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777059.png" />, the associated Stiefel bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777060.png" /> have the fibres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777061.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777062.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777063.png" /> is the fibre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777064.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777065.png" />. Similarly one has the Grassmann bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777066.png" />, whose fibre over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777067.png" /> is the Grassmann manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777068.png" />.
dimensional (real, complex, quaternionic) [[Vector bundle|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====
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777069.png" />"  ''Quarterly J. Math.'' , '''7'''  (1956)  pp. 249–268</TD></TR><TR><TD valign="top">[a3b]</TD> <TD valign="top">  G.F. Paechter,  "The groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777070.png" />"  ''Quarterly J. Math.'' , '''9'''  (1958)  pp. 8–27</TD></TR><TR><TD valign="top">[a3c]</TD> <TD valign="top">  G.F. Paechter,  "The groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777071.png" />"  ''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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777072.png" />"  ''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>
 
<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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777069.png" />"  ''Quarterly J. Math.'' , '''7'''  (1956)  pp. 249–268</TD></TR><TR><TD valign="top">[a3b]</TD> <TD valign="top">  G.F. Paechter,  "The groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777070.png" />"  ''Quarterly J. Math.'' , '''9'''  (1958)  pp. 8–27</TD></TR><TR><TD valign="top">[a3c]</TD> <TD valign="top">  G.F. Paechter,  "The groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777071.png" />"  ''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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087770/s08777072.png" />"  ''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>

Revision as of 14:53, 7 June 2020

(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)
How to Cite This Entry:
Stiefel manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stiefel_manifold&oldid=48838
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article