Grassmann manifolds

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- For references, use e.g. {{cite|Milnor1958b}}.
- For references, use e.g. {{cite|Milnor1958b}}.
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== Introduction ==
== Introduction ==
<wikitex>;
<wikitex>;
Grassmann manifolds are named after Hermann Grassmann, a German school teacher in Stettin who developed basic notions of linear algebra. Let $K$ be the real, complex or quaternion field and $V$ a vector space over $K$. 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. Let $K$ be the real, complex or quaternion field and $V$ a vector space over $K$ 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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{{beginthm|Theorem|{{cite|{Milnor&Stasheff1974}}}}} $G_k(V)$ is a Hausdorff, compact space. {{endthm}}
{{beginthm|Theorem|{{cite|}}}} $G_k(V)$ is a Hausdorff, compact space. \end{zad}
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{{beginthm|Theorem|{{cite|{Milnor&Stasheff1974}}}}} $G_k(V)$ is a connected, compact smooth manifold of dimension $dk(n-k)$. {{endthm}}
\begin{zad} Prove that there exist a homeomorphism $G_k(\F^n)\simeq G_{n-k}(\F^n)$ \end{zad}
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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 $Hom (W^\bot ,W).$ {{endthm}}
\begin{zad} Prove that $G_k(\F^n)$ is a connected, compact manifold of dimension $dk(n-k)$. \end{zad}
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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 $\O(W^\bot)\times O(W)$.
\begin{zad} 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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{{beginthm|Proposition|{{cite|{Milnor&Stasheff1974}}}}} There exist a natural diffeomorphism $G_k(V)\simeq G_{n-k}(V^*)$ {{endthm}}
\end{zad}
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The Grassmannians $G_1(V)$ are the well-known projective spaces, denoted $P(V)$. Note that $G_1(F^2)=S^d$ and if we identify $S^d$ with the one-point compactification of $\F$ the projection $p$ 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 $\F=\OO$, 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 $S^{d-1}$. Check directly that the Hopf maps are locally trivial, thus fibrations.
\begin{zad} The Grassmannians $G_1(\F^n)$ are the well-known projective spaces, denoted $\F P(n)$. Note that $G_1(\F^2)=S^d$ and if we identify $S^d$ with the one-point compactification of $\F$ the projection $p$ 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 $\F=\OO$, 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 $S^{d-1}$. Check directly that the Hopf maps are locally trivial, thus fibrations. \end{zad}
\begin{zad} The natural action of $GL(n,\F)$ (resp. $O(n,\F)$) on $\F^n$ induces an action on $G_k(\F^n)$. Show that the actions are transitive and describe the isotropy groups (in particular of the canonical subspace $F^k\subset F^n$)
\end{zad}
\begin{zad} 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.
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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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+
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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}
\end{zad}
\begin{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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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.
\end{zad}
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They are examples of coadjoint orbits \cite{Kirillov2004}
They are examples of coadjoint orbits \cite{Kirillov2004}

Revision as of 12:26, 26 November 2010


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Contents

1 Introduction

Grassmann manifolds are named after Hermann Grassmann, a German school teacher in Stettin who developed basic notions of linear algebra. Let K be the real, complex or quaternion field and V a vector space over K 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.

Theorem 1.1 [{Milnor&Stasheff1974].} G_k(V) is a Hausdorff, compact space.
Theorem 1.2 [{Milnor&Stasheff1974].} G_k(V) is a connected, compact smooth manifold of dimension dk(n-k).

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). </div>

Grassmann manifold is a homogeneous space of the general linear group. General linear group
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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).
Proposition 1.3 [{Milnor&Stasheff1974].} There exist a natural diffeomorphism G_k(V)\simeq G_{n-k}(V^*)
The Grassmannians G_1(V) are the well-known projective spaces, denoted P(V). Note that G_1(F^2)=S^d and if we identify S^d with the one-point compactification of \F the projection p 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
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, however the higher dimensional projective spaces over octonions do not exist. The maps p_d :S^{2d-1}\to S^d for
Tex syntax error
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}. Check directly that the Hopf maps are locally trivial, thus fibrations.



There is an embedding of the Grassmannian
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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
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.


Prove that there is a free action of the group
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on
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sucht that the orbit space is homeomorphic to
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. Similarly for the noncompact Stiefel manifold.

\end{zad}

Prove that the map
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is locally trivial (even a principal
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-bundle), thus a fibration.

They are examples of coadjoint orbits [Kirillov2004]

[Milnor&Stasheff1974]

Theorem 1.4.



2 Construction and examples

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3 Invariants

...

4 Classification/Characterization

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5 Further discussion

...

6 References

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