Stiefel manifold

 

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In mathematics , the Stiefel manifold V k (R n ) is the set of all orthonormal k -frames in R n . That is, it is the set of ordered k -tuples of orthonormal vectors in R n . It is named after Swiss mathematician Eduard Stiefel . Likewise one can define the complex Stiefel manifold V k (C n ) of orthonormal k -frames in C n and the quaternionic Stiefel manifold V k (H n ) of orthonormal k -frames in H n . More generally, the construction applies to any real, complex, or quaternionic inner product space .

In some contexts, a non-compact Stiefel manifold is defined as the set of all linearly independent k -frames in R n , C n , or H n ; this is homotopy equivalent, as the compact Stiefel manifold is a deformation retract of the non-compact one, by Gram–Schmidt . Statements about the non-compact form correspond to those for the compact form, replacing the orthogonal group (or unitary or symplectic group) with the general linear group .

Contents

[hide ]
  • 1 Topology
  • 2 As a homogeneous space
  • 3 Special cases
  • 4 As a principal bundle
  • 5 Homotopy
  • 6 See also
  • 7 References
  • 8 External links

[edit ] Topology

Let F stand for R , C , or H . The Stiefel manifold V k (F n ) can be thought of as a set of n × k matrices by writing a k -frame as a matrix of k column vectors in F n . The orthonormality condition is expressed by A *A = 1 where A * denotes the conjugate transpose of A and 1 denotes the k × k identity matrix . We then have

V_k(/mathbb F^n) = /left/{A /in /mathbb F^{n/times k} : A^{/ast}A = 1/right/}.

The topology on V k (F n ) is the subspace topology inherited from F n ×k . With this topology V k (F n ) is a compact manifold whose dimension is given by

/dim V_k(/mathbb R^n) = nk - /frac{1}{2}k(k+1)
/dim V_k(/mathbb C^n) = 2nk - k^2
/dim V_k(/mathbb H^n) = 4nk - k(2k-1).

[edit ] As a homogeneous space

Each of the Stiefel manifolds V k (F n ) can be viewed as a homogeneous space for the action of a classical group in a natural manner.

Every orthogonal transformation of a k -frame in R n results in another k -frame, and any two k -frames are related by some orthogonal transformation. In other words, the orthogonal group O(n ) acts transitively on V k (R n ). The stabilizer subgroup of a given frame is the subgroup isomorphic to O(nk ) which acts nontrivially on the orthogonal complement of the space spanned by that frame.

Likewise the unitary group U(n ) acts transitively on V k (C n ) with stabilizer subgroup U(nk ) and the symplectic group Sp(n ) acts transitively on V k (H n ) with stabilizer subgroup Sp(nk ).

In each case V k (F n ) can be viewed as a homogeneous space:

/begin{align} V_k(/mathbb R^n) &/cong /mbox{O}(n)//mbox{O}(n-k)// V_k(/mathbb C^n) &/cong /mbox{U}(n)//mbox{U}(n-k)// V_k(/mathbb H^n) &/cong /mbox{Sp}(n)//mbox{Sp}(n-k). /end{align}

When k = n , the corresponding action is free so that the Stiefel manifold V n (F n ) is a principal homogeneous space for the corresponding classical group.

When k is strictly less than n then the special orthogonal group SO(n ) also acts transitively on V k (R n ) with stabilizer subgroup isomorphic to SO(nk ) so that

V_k(/mathbb R^n) /cong /mbox{SO}(n)//mbox{SO}(n-k)/qquad/mbox{for } k < n.

The same holds for the action of the special unitary group on V k (C n )

V_k(/mathbb C^n) /cong /mbox{SU}(n)//mbox{SU}(n-k)/qquad/mbox{for } k < n.

Thus for k = n − 1, the Stiefel manifold is a principal homogeneous space for the corresponding special classical group.

[edit ] Special cases

k = 1 /begin{align} V_1(/mathbb R^n) &= S^{n-1}// V_1(/mathbb C^n) &= S^{2n-1}// V_1(/mathbb H^n) &= S^{4n-1} /end{align}
k = n −1 /begin{align} V_{n-1}(/mathbb R^n) &/cong /mathrm{SO}(n)// V_{n-1}(/mathbb C^n) &/cong /mathrm{SU}(n) /end{align}
k = n /begin{align} V_{n}(/mathbb R^n) &/cong /mathrm O(n)// V_{n}(/mathbb C^n) &/cong /mathrm U(n)// V_{n}(/mathbb H^n) &/cong /mathrm{Sp}(n) /end{align}

A 1-frame in F n is nothing but a unit vector, so the Stiefel manifold V 1 (F n ) is just the unit sphere in F n .

Given a 2-frame in R n , let the first vector define a point in S n −1 and the second a unit tangent vector to the sphere at that point. In this way, the Stiefel manifold V 2 (R n ) may be identified with the unit tangent bundle to S n −1 .

When k = n or n −1 we saw in the previous section that V k (F n ) is a principal homogeneous space, and therefore diffeomorphic to the corresponding classical group. These are listed in the table at the right.

[edit ] As a principal bundle

There is a natural projection

p: V_k(/mathbb F^n) /to G_k(/mathbb F^n)

from the Stiefel manifold V k (F n ) to the Grassmannian of k -planes in F n which sends a k -frame to the subspace spanned by that frame. The fiber over a given point P in G k (F n ) is the set of all orthonormal k -frames contained in the space P .

This projection has the structure of a principal G -bundle where G is the associated classical group of degree k . Take the real case for concreteness. There is a natural right action of O(k ) on V k (R n ) which rotates a k -frame in the space it spans. This action is free but not transitive. The orbits of this action are precisely the orthonormal k -frames spanning a given k -dimensional subspace; that is, they are the fibers of the map p . Similar arguments hold in the complex and quaternionic cases.

We then have a sequence of principal bundles:

/begin{align} /mathrm O(k) &/to V_k(/mathbb R^n) /to G_k(/mathbb R^n)// /mathrm U(k) &/to V_k(/mathbb C^n) /to G_k(/mathbb C^n)// /mathrm{Sp}(k) &/to V_k(/mathbb H^n) /to G_k(/mathbb H^n). /end{align}

The vector bundles associated to these principal bundles via the natural action of G on F k are just the tautological bundles over the Grassmannians. In other words, the Stiefel manifold V k (F n ) is the orthogonal, unitary, or symplectic frame bundle associated to the tautological bundle on a Grassmannian.

When one passes to the n → ∞ limit, these bundles become the universal bundles for the classical groups.

[edit ] Homotopy

The Stiefel manifolds fit into a family of fibrations V_{k-1}(/Bbb R^{n-1}) /to V_k(/Bbb R^n) /to S^{n-1} , thus the first non-trivial homotopy group of the space V_k(/Bbb R^n) is in dimension nk . Moreover, /pi_{n-k} V_k(/Bbb R^n) /simeq /Bbb Z if n-k /in 2 /Bbb Z or if k = 1 . /pi_{n-k} V_k(/Bbb R^n) /simeq /Bbb Z_2 if nk is odd and k > 1 . This result is used in the obstruction-theoretic definition of Stiefel-Whitney classes .

[edit ] See also

  • Flag manifold

[edit ] References

  • Hatcher, Allen (2002). Algebraic Topology . Cambridge University Press. ISBN  0-521-79540-0 .
  • Husemoller, Dale (1994). Fibre Bundles ((3rd ed.) ed.). New York: Springer-Verlag. ISBN  0-387-94087-1 .
  • James, Ioan Mackenzie (1976). The topology of Stiefel manifolds . CUP Archive. ISBN  978-0-52121334-9 .

[edit ] External links

  • Encyclopaedia of Mathematics » Stiefel manifold , Springer

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