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

How to Cite This Entry:
Stiefel manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stiefel_manifold&oldid=53839
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article