Grassmann manifolds

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== Introduction ==
== Introduction ==
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Grassmann manifolds are named after Hermann Grassmann, a German school teacher in Stettin who developed basic notions of linear algebra. Let $\F$ be the real, complex or quaternion field and $V$ a vector space over $F$ of dimension $n$ and let $k\leq n$. A Grassmann manifolds of $k$-dimensional subspaces is a set $G_k(V)$ of $k$-dimensional subspaces. The set $G_k(V)$ is a quotient of a subset of $V\times ...\times V$ consisting of linearly independent $k$-tuples of vectors with the subspace topology. We define topology on $G_k(V)$ as the quotient topology.
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Grassmann manifolds are named after Hermann Grassmann, a German school teacher in Stettin who developed basic notions of linear algebra. They play a key role in topology and geometry as the universal spaces of vector bundles. See also [[Wikipedia:Grassmannian|Grassmannian]]
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{{beginthm|Theorem|{{cite|Milnor&Stasheff1974}}}} $G_k(V)$ is a Hausdorff, compact space. {{endthm}}
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== Construction and examples ==
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{{beginthm|Theorem|{{cite|Milnor&Stasheff1974}}}} $G_k(V)$ is a connected, compact smooth manifold of dimension $dk(n-k)$. {{endthm}}
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=== Construction ===
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Let $\F=\Rr ,\Cc , \Hh$ be the real, complex or quaternion field and $V$ a vector space over $\F$ of dimension $n$ and let $k\leq n$. A Grassmannian of $k$-dimensional subspaces is a set $G_k(V)$ of $k$-dimensional subspaces. The set $G_k(V)$ is a quotient of a subset of $V\times ...\times V$ consisting of linearly independent $k$-tuples of vectors with the subspace topology. We define topology on $G_k(V)$ as the quotient topology. Grassmannian is a homogeneous space of the general linear group. General linear group $\GL(V)$ acts transitively on $G_k(V)$ with an isotropy group consisting of automorphisms preserving a given subspace. If the space $V$ is equipped with a scalar product (hermitian metric resp.) then the group of isometries $O(V)$ acts transitively and the isotropy group of $W$ is $O(W^\bot)\times O(W)$.
Note that the Grassmann manifold $G_k(V)$ around $W\in G_k(V)$ is locally modelled on the vector space $Hom (W^\bot ,W).$ {{endthm}}
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{{beginthm|Theorem|{{cite|Milnor&Stasheff1974}}}} $G_k(V)$ is a Hausdorff, compact, connected smooth manifold of dimension $dk(n-k)$. For $\F=\Cc ,\Hh$ it is also a complex manifold.{{endthm}}
Grassmann manifold is a homogeneous space of the general linear group. General linear group $GL(V)$ acts transitively on $G_k(V)$ with an isotropy group consisting of automorphisms preserving a given subspace. If the space $V$ is equipped with a scalar product (hermitian metric resp.) then the group of isometries $O(V)$ acts transitively and the isotropy group of $W$ is $O(W^\bot)\times O(W)$.
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Note that the Grassmann manifold $G_k(V)$ around $W\in G_k(V)$ is locally modelled on the vector space $Hom (W^\bot ,W).$
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{{beginthm|Proposition|}} There exist a natural diffeomorphism $G_k(V)\simeq G_{n-k}(V^*)$. {{endthm}}
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=== The canonical bundle ===
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The Grassmann manifold is equipped with the canonical, tautological vector bundle $\gamma^V_k.$ which is a subbundle of the trivial bundle $G_k(V)\times V \to G_k(V)\times V$. The total space is $E(\gamma^V_k) = \{(W,w)\in G_k(V)\times V\, |\,\, w\in W \}$ The total space of the associated principal bundle is a Stiefel manifold.
The Grassmann manifold is equipped with the canonical, tautological vector bundle $\gamma^V_k.$ which is a subbundle of the trivial bundle $G_k(V)\times V \to G_k(V)\times V$. The total space is $E(\gamma^V_k) = \{(W,w)\in G_k(V)\times V\, |\,\, w\in W \}$ The total space of the associated principal bundle is a Stiefel manifold.
{{beginthm|Proposition|{{cite|Milnor&Stasheff1974}}}} There exist a natural diffeomorphism $G_k(V)\simeq G_{n-k}(V^*)$. {{endthm}}
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The tangent bundle to Grassmaniann can be expressed in terms of the canonical bundle: $TG_k(V)= \Hom (\gamma_k^\bot , \gamma_k).$</wikitex>
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=== Low dimensional Grassmannians ===
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The Grassmannians $G_1(V)$ are projective spaces, denoted $P (V)$. Note that $G_1(F^2)=S^d$, where $d=dim_{\Rr} F$. If we identify $S^d$ with the one-point compactification of $\F$ the projection of the canonical principal bundle corresponds to the map $p_d :S^{2d-1}\to S^d$ given by $p_d(z_0,z_1)=z_0/z_1$ where $z_i\in\F$. Note, that the same formula works for octonions $\Oo$, however the higher dimensional projective spaces over octonions do not exist. The maps $p_d :S^{2d-1}\to S^d$ for $d= 2,4,8$ are called the Hopf maps and they play a very important role in homotopy theory; a fibre of $p_d$ is a sphere $S^{d-1}$.
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=== Embeddings of Grassmannians into affine and projective space ===
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There is an embedding of the Grassmannian $G_k(V)$ in the Cartesian space $\F^{n^2}=\Hom\,(F^n,F^n)$ which assigns to every subspace the orthogonal projection on it. If $V$ is equipped with a norm, the embedding defines a natural (operator) metric on $G_k(V)$.
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=== Infinite dimensional Grassmannians ===
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Infinite dimensional Grassmannians. Natural inclusions of vector space $\F ^1 \subset \F ^2 \subset ...\F ^n \subset ...$ defines inclusions of Grassmannians. The colimit of the resulting sequence is denoted $G_k(\F^{\infty} )$ and also $BGL(k,\F)$. One can also take the colimit with respect to both dimension of the space and of the subspaces. We have a sequence of inclusions $G_1(\F^2)\subset G_2(\F^4)\subset ... \subset G_n(\F^{2n}) \subset ...$ and its colimit is denoted $B\GL (\F).$
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</wikitex>
The Grassmannians $G_1(V)$ are projective spaces, denoted $P (V)$. Note that $G_1(F^2)=S^d$, where $d=dim_R F$. If we identify $S^d$ with the one-point compactification of $\F$ the projection of the canonical principal bundle corresponds to the map $p_d :S^{2d-1}\to S^d$ given by $p_d(z_0,z_1)=z_0/z_1$ where $z_i\in\F$. Note, that the same formula works for octonions, however the higher dimensional projective spaces over octonions do not exist. The maps $p_d :S^{2d-1}\to S^d$ for $d=1,2,4,8$ are called the Hopf maps and they play a very important role in homotopy theory; a fiber of $p_d$ is a sphere $S^{d-1}$.
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== Invariants ==
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There is an embedding of the Grassmannian $G_k(\F^n)$ in the cartesian space $\F^{n^2}=\Hom\,(F^n,F^n)$ which assigns to every subsapce the orthogonal projection on it. The embedding defines a natural (operator) metric on $G_k(\F^n)$.
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Prove that there is a free action of the group $O(k,\F)$ on $V_k(\F^n)$ sucht that the orbit space is homeomorphic to $G_k(\F^n)$. Similarly for the noncompact Stiefel manifold.
\end{zad}
Prove that the map $p:V_k(\F^n)\to G_k(\F^n)$ is locally trivial (even a principal $O(k,\F)$-bundle), thus a fibration.
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=== Homotopy groups ===
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Homotopy groups of Grassmannians are closely related to homotopy groups of spheres via the appropriate fibration sequences. They also imply that the groups $\pi_i(G_k (V))$ do not depend on $V$, if $k\leq\leq dim V.$ Homotopy groups in the stable range are described by the Bott periodicity theorem.
== Construction and examples ==
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{{beginthm|Proposition|{(R.Bott)}}} For each $i>0$ there are isomorphisms: $\pi_i(BGL(\Rr)) \simeq \pi_{i+8}(BGL(\Rr))$ and $\pi_i(BGL(\Cc)) \simeq \pi_{i+2}(BGL(\Cc))$
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{{endthm}}
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== Invariants ==
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=== Cohomology groups ===
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== Classification/Characterization ==
== Classification/Characterization ==
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== Further discussion ==
== Further discussion ==
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Grassmann manifolds are examples of [[coadjoint orbits]] \cite{Kirillov2004}.
Grassmann manifolds are examples of [[coadjoint orbits]] \cite{Kirillov2004}.
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== References ==
== References ==
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==External links==
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* The Wikipedia page on [[Wikipedia:Grassmannian|Grassmannian]]
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[[Category:Manifolds]]
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Latest revision as of 18:05, 12 October 2013

This page has not been refereed. The information given here might be incomplete or provisional.

Contents

[edit] 1 Introduction

Grassmann manifolds are named after Hermann Grassmann, a German school teacher in Stettin who developed basic notions of linear algebra. They play a key role in topology and geometry as the universal spaces of vector bundles. See also Grassmannian

[edit] 2 Construction and examples

[edit] 2.1 Construction

Let \F=\Rr ,\Cc , \Hh be the real, complex or quaternion field and V a vector space over
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of dimension n and let
Tex syntax error
. A Grassmannian of k-dimensional subspaces is a set
Tex syntax error
of k-dimensional subspaces. The set
Tex syntax error
is a quotient of a subset of
Tex syntax error
consisting of linearly independent k-tuples of vectors with the subspace topology. We define topology on
Tex syntax error
as the quotient topology. Grassmannian is a homogeneous space of the general linear group. General linear group \GL(V) acts transitively on
Tex syntax error
with an isotropy group consisting of automorphisms preserving a given subspace. If the space V is equipped with a scalar product (hermitian metric resp.) then the group of isometries
Tex syntax error
acts transitively and the isotropy group of
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is
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.
Theorem 2.1 [Milnor&Stasheff1974].
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is a Hausdorff, compact, connected smooth manifold of dimension
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. For \F=\Cc ,\Hh it is also a complex manifold.
Note that the Grassmann manifold
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around
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is locally modelled on the vector space
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Proposition 2.2. There exist a natural diffeomorphism
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.

[edit] 2.2 The canonical bundle

The Grassmann manifold is equipped with the canonical, tautological vector bundle
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which is a subbundle of the trivial bundle
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. The total space is
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The total space of the associated principal bundle is a Stiefel manifold.

The tangent bundle to Grassmaniann can be expressed in terms of the canonical bundle: TG_k(V)= \Hom (\gamma_k^\bot , \gamma_k).

[edit] 2.3 Low dimensional Grassmannians

The Grassmannians
Tex syntax error
are projective spaces, denoted
Tex syntax error
. Note that
Tex syntax error
, where d=dim_{\Rr} F. If we identify
Tex syntax error
with the one-point compactification of
Tex syntax error
the projection of the canonical principal bundle corresponds to the map
Tex syntax error
given by
Tex syntax error
where
Tex syntax error
. Note, that the same formula works for octonions \Oo, however the higher dimensional projective spaces over octonions do not exist. The maps
Tex syntax error
for d= 2,4,8 are called the Hopf maps and they play a very important role in homotopy theory; a fibre of
Tex syntax error
is a sphere
Tex syntax error
.

[edit] 2.4 Embeddings of Grassmannians into affine and projective space

There is an embedding of the Grassmannian
Tex syntax error
in the Cartesian space
Tex syntax error
which assigns to every subspace the orthogonal projection on it. If V is equipped with a norm, the embedding defines a natural (operator) metric on
Tex syntax error
.

[edit] 2.5 Infinite dimensional Grassmannians

Infinite dimensional Grassmannians. Natural inclusions of vector space \F ^1 \subset \F ^2 \subset ...\F ^n \subset ... defines inclusions of Grassmannians. The colimit of the resulting sequence is denoted G_k(\F^{\infty} ) and also BGL(k,\F). One can also take the colimit with respect to both dimension of the space and of the subspaces. We have a sequence of inclusions G_1(\F^2)\subset G_2(\F^4)\subset ... \subset G_n(\F^{2n}) \subset ... and its colimit is denoted B\GL (\F).

[edit] 3 Invariants


[edit] 3.1 Homotopy groups

Homotopy groups of Grassmannians are closely related to homotopy groups of spheres via the appropriate fibration sequences. They also imply that the groups \pi_i(G_k (V)) do not depend on V, if k\leq\leq dim V. Homotopy groups in the stable range are described by the Bott periodicity theorem.

Proposition 3.1 {(R.Bott).} For each i>0 there are isomorphisms: \pi_i(BGL(\Rr)) \simeq \pi_{i+8}(BGL(\Rr)) and \pi_i(BGL(\Cc)) \simeq \pi_{i+2}(BGL(\Cc))

[edit] 3.2 Cohomology groups

...

[edit] 4 Classification/Characterization

...

[edit] 5 Further discussion

Grassmann manifolds are examples of coadjoint orbits [Kirillov2004].

[edit] 6 References

[edit] 7 External links

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