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| In [[mathematics]], the '''Stiefel manifold''' ''V''<sub>''k''</sub>('''R'''<sup>''n''</sup>) is the set of all [[orthonormal]] [[k-frame|''k''-frames]] in '''R'''<sup>''n''</sup>. That is, it is the set of ordered ''k''-tuples of orthonormal [[vector (mathematics)|vectors]] in '''R'''<sup>''n''</sup>. It is named after Swiss mathematician [[Eduard Stiefel]]. Likewise one can define the [[complex number|complex]] Stiefel manifold ''V''<sub>''k''</sub>('''C'''<sup>''n''</sup>) of orthonormal ''k''-frames in '''C'''<sup>''n''</sup> and the [[quaternion]]ic Stiefel manifold ''V''<sub>''k''</sub>('''H'''<sup>''n''</sup>) of orthonormal ''k''-frames in '''H'''<sup>''n''</sup>. More generally, the construction applies to any real, complex, or quaternionic [[inner product space]].
| | Irwin Butts is what my wife loves to call me although I don't really like being known as like that. Puerto Rico is exactly where he and his wife live. My working day job is a librarian. To collect coins is one of the things I love most.<br><br>Also visit my web page; [http://ucmasuae.com/members/nellschauer/activity/43826/ home std test] |
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| In some contexts, a non-[[compact space|compact]] Stiefel manifold is defined as the set of all [[linearly independent]] ''k''-frames in '''R'''<sup>''n''</sup>, '''C'''<sup>''n''</sup>, or '''H'''<sup>''n''</sup>; this is homotopy equivalent, as the compact Stiefel manifold is a [[deformation retract]] of the non-compact one, by [[Gram–Schmidt process|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]].
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| ==Topology==
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| Let '''F''' stand for '''R''', '''C''', or '''H'''. The Stiefel manifold ''V''<sub>''k''</sub>('''F'''<sup>''n''</sup>) can be thought of as a set of ''n'' × ''k'' [[matrix (mathematics)|matrices]] by writing a ''k''-frame as a matrix of ''k'' [[column vector]]s in '''F'''<sup>''n''</sup>. 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
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| :<math>V_k(\mathbb F^n) = \left\{A \in \mathbb F^{n\times k} : A^{\ast}A = 1\right\}.</math>
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| The [[topology (structure)|topology]] on ''V''<sub>''k''</sub>('''F'''<sup>''n''</sup>) is the [[subspace topology]] inherited from '''F'''<sup>''n''×''k''</sup>. With this topology ''V''<sub>''k''</sub>('''F'''<sup>''n''</sup>) is a [[compact space|compact]] [[manifold]] whose dimension is given by
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| :<math>\dim V_k(\mathbb R^n) = nk - \frac{1}{2}k(k+1)</math>
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| :<math>\dim V_k(\mathbb C^n) = 2nk - k^2</math>
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| :<math>\dim V_k(\mathbb H^n) = 4nk - k(2k-1).</math>
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| ==As a homogeneous space==
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| Each of the Stiefel manifolds ''V''<sub>''k''</sub>('''F'''<sup>''n''</sup>) can be viewed as a [[homogeneous space]] for the [[group action|action]] of a [[classical group]] in a natural manner.
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| Every orthogonal transformation of a ''k''-frame in '''R'''<sup>''n''</sup> 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 [[transitive action|transitively]] on ''V''<sub>''k''</sub>('''R'''<sup>''n''</sup>). The [[stabilizer subgroup]] of a given frame is the subgroup isomorphic to O(''n''−''k'') which acts nontrivially on the [[orthogonal complement]] of the space spanned by that frame.
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| Likewise the [[unitary group]] U(''n'') acts transitively on ''V''<sub>''k''</sub>('''C'''<sup>''n''</sup>) with stabilizer subgroup U(''n''−''k'') and the [[symplectic group]] Sp(''n'') acts transitively on ''V''<sub>''k''</sub>('''H'''<sup>''n''</sup>) with stabilizer subgroup Sp(''n''−''k'').
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| In each case ''V''<sub>''k''</sub>('''F'''<sup>''n''</sup>) can be viewed as a homogeneous space:
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| :<math>\begin{align}
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| V_k(\mathbb R^n) &\cong \mbox{O}(n)/\mbox{O}(n-k)\\
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| V_k(\mathbb C^n) &\cong \mbox{U}(n)/\mbox{U}(n-k)\\
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| V_k(\mathbb H^n) &\cong \mbox{Sp}(n)/\mbox{Sp}(n-k).
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| \end{align}</math>
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| When ''k'' = ''n'', the corresponding action is free so that the Stiefel manifold ''V''<sub>''n''</sub>('''F'''<sup>''n''</sup>) is a [[principal homogeneous space]] for the corresponding classical group.
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| When ''k'' is strictly less than ''n'' then the [[special orthogonal group]] SO(''n'') also acts transitively on ''V''<sub>''k''</sub>('''R'''<sup>''n''</sup>) with stabilizer subgroup isomorphic to SO(''n''−''k'') so that
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| :<math>V_k(\mathbb R^n) \cong \mbox{SO}(n)/\mbox{SO}(n-k)\qquad\mbox{for } k < n.</math>
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| The same holds for the action of the [[special unitary group]] on ''V''<sub>''k''</sub>('''C'''<sup>''n''</sup>)
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| :<math>V_k(\mathbb C^n) \cong \mbox{SU}(n)/\mbox{SU}(n-k)\qquad\mbox{for } k < n.</math>
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| Thus for ''k'' = ''n'' - 1, the Stiefel manifold is a principal homogeneous space for the corresponding ''special'' classical group.
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| ==Special cases==
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| {| class="wikitable" style="float:right; background:white; margin:0em 0em 0em 1em;"
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| |-
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| |style="text-align:center;"|''k'' = 1
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| |<math>\begin{align}
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| V_1(\mathbb R^n) &= S^{n-1}\\
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| V_1(\mathbb C^n) &= S^{2n-1}\\
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| V_1(\mathbb H^n) &= S^{4n-1}
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| \end{align}</math>
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| |-
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| |style="text-align:center;"|''k'' = ''n''−1
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| |<math>\begin{align}
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| V_{n-1}(\mathbb R^n) &\cong \mathrm{SO}(n)\\
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| V_{n-1}(\mathbb C^n) &\cong \mathrm{SU}(n)
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| \end{align}</math>
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| |-
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| |style="text-align:center;"|''k'' = ''n''
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| |<math>\begin{align}
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| V_{n}(\mathbb R^n) &\cong \mathrm O(n)\\
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| V_{n}(\mathbb C^n) &\cong \mathrm U(n)\\
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| V_{n}(\mathbb H^n) &\cong \mathrm{Sp}(n)
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| \end{align}</math>
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| |}
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| A 1-frame in '''F'''<sup>''n''</sup> is nothing but a unit vector, so the Stiefel manifold ''V''<sub>1</sub>('''F'''<sup>''n''</sup>) is just the [[unit sphere]] in '''F'''<sup>''n''</sup>.
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| Given a 2-frame in '''R'''<sup>''n''</sup>, let the first vector define a point in ''S''<sup>''n''−1</sup> and the second a unit [[tangent vector]] to the sphere at that point. In this way, the Stiefel manifold ''V''<sub>2</sub>('''R'''<sup>''n''</sup>) may be identified with the [[unit tangent bundle]] to ''S''<sup>''n''−1</sup>.
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| When ''k'' = ''n'' or ''n''−1 we saw in the previous section that ''V''<sub>''k''</sub>('''F'''<sup>''n''</sup>) is a principal homogeneous space, and therefore [[diffeomorphic]] to the corresponding classical group. These are listed in the table at the right.
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| <div style="clear:both;"></div>
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| ==Functoriality==
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| Given an orthogonal inclusion between vector spaces <math>X \hookrightarrow Y,</math> the image of a set of ''k'' orthonormal vectors is orthonormal, so there is an induced closed inclusion of Stiefel manifolds, <math>V_k(X) \hookrightarrow V_k(Y),</math> and this is [[functorial]]. More subtly, given an ''n''-dimensional vector space ''X,'' the [[dual basis]] construction gives a bijection between bases for ''X'' and bases for the dual space ''X''<sup>*</sup>, which is continuous, and thus yields a homeomorphism of top Stiefel manifolds <math>V_n(X) \stackrel{\sim}{\to} V_n(X^*).</math> This is also functorial for isomorphisms of vector spaces.
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| ==As a principal bundle==
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| There is a natural projection
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| :<math>p: V_k(\mathbb F^n) \to G_k(\mathbb F^n)</math>
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| from the Stiefel manifold ''V''<sub>''k''</sub>('''F'''<sup>''n''</sup>) to the [[Grassmannian]] of ''k''-planes in '''F'''<sup>''n''</sup> which sends a ''k''-frame to the [[linear subspace|subspace]] spanned by that frame. The [[fiber (mathematics)|fiber]] over a given point ''P'' in ''G''<sub>''k''</sub>('''F'''<sup>''n''</sup>) is the set of all orthonormal ''k''-frames contained in the space ''P''.
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| This projection has the structure of a [[principal bundle|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''<sub>''k''</sub>('''R'''<sup>''n''</sup>) which rotates a ''k''-frame in the space it spans. This action is free but not transitive. The [[orbit (group theory)|orbit]]s 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.
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| We then have a sequence of principal bundles:
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| :<math>\begin{align}
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| \mathrm O(k) &\to V_k(\mathbb R^n) \to G_k(\mathbb R^n)\\
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| \mathrm U(k) &\to V_k(\mathbb C^n) \to G_k(\mathbb C^n)\\
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| \mathrm{Sp}(k) &\to V_k(\mathbb H^n) \to G_k(\mathbb H^n).
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| \end{align}</math>
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| The [[vector bundle]]s [[associated bundle|associated]] to these principal bundles via the natural action of ''G'' on '''F'''<sup>''k''</sup> are just the [[tautological bundle]]s over the Grassmannians. In other words, the Stiefel manifold ''V''<sub>''k''</sub>('''F'''<sup>''n''</sup>) is the orthogonal, unitary, or symplectic [[frame bundle]] associated to the tautological bundle on a Grassmannian.
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| When one passes to the ''n'' → ∞ limit, these bundles become the [[universal bundle]]s for the classical groups.
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| ==Homotopy==
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| The Stiefel manifolds fit into a family of [[Serre fibration|fibrations]] <math>V_{k-1}(\Bbb R^{n-1}) \to V_k(\Bbb R^n) \to S^{n-1}</math>, thus the first non-trivial [[homotopy group]] of the space ''V''<sub>''k''</sub>('''R'''<sup>''n''</sup>) is in dimension ''n'' - ''k''. Moreover, <math>\pi_{n-k} V_k(\Bbb R^n) \simeq \Bbb Z</math> if ''n'' - ''k'' ∈ 2'''Z''' or if ''k'' = 1. <math>\pi_{n-k} V_k(\Bbb R^n) \simeq \Bbb Z_2</math> if ''n'' - ''k'' is odd and ''k'' > 1. This result is used in the obstruction-theoretic definition of [[Stiefel-Whitney class]]es.
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| ==See also==
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| *[[Flag manifold]]
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| ==References==
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| *{{cite book | first = Allen | last = Hatcher | authorlink = Allen Hatcher | year = 2002 | title = Algebraic Topology | publisher = Cambridge University Press | isbn = 0-521-79540-0 | url = http://www.math.cornell.edu/~hatcher/AT/ATpage.html}}
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| *{{cite book | first = Dale | last = Husemoller | year = 1994 | title = Fibre Bundles | edition = (3rd ed.) | publisher = Springer-Verlag | location = New York | isbn = 0-387-94087-1}}
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| *{{cite book | first = Ioan Mackenzie | last = James | year = 1976 | title = The topology of Stiefel manifolds | publisher = CUP Archive | isbn = 978-0-521-21334-9 | url = http://books.google.com/books?id=9ss7AAAAIAAJ}}
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| ==External links==
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| * Encyclopaedia of Mathematics » [http://www.encyclopediaofmath.org/index.php/Stiefel_manifold Stiefel manifold], Springer
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| [[Category:Differential geometry]]
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| [[Category:Homogeneous spaces]]
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| [[Category:Fiber bundles]]
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| [[Category:Manifolds]]
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Irwin Butts is what my wife loves to call me although I don't really like being known as like that. Puerto Rico is exactly where he and his wife live. My working day job is a librarian. To collect coins is one of the things I love most.
Also visit my web page; home std test