# 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

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,

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

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

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

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:

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

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 " 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=49604
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article