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 ==
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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 $K$. 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.	+	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.
		+	{{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}	+	{{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}	+	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}	+	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 $G_k(\F^n)$.	+	{{beginthm|Proposition|{{cite|{Milnor&Stasheff1974}}}}} There exist a natural diffeomorphism $G_k(V)\simeq G_{n-k}(V^*)$ {{endthm}}
−	\end{zad}	+
		+	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.	+	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 $G_k(\F^n)$.
		+
		+
		+	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.	+	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}	+
	They are examples of coadjoint orbits \cite{Kirillov2004}		They are examples of coadjoint orbits \cite{Kirillov2004}

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Revision as of 13:26, 26 November 2010

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Contents 1 Introduction 2 Construction and examples 3 Invariants 4 Classification/Characterization 5 Further discussion 6 References

1 Introduction

Grassmann manifolds are named after Hermann Grassmann, a German school teacher in Stettin who developed basic notions of linear algebra. Let $<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> {{Stub}} == Introduction == <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. {{beginthm|Theorem|{{cite|}}}} $G_k(V)$ is a Hausdorff, compact space. \end{zad} \begin{zad} Prove that there exist a homeomorphism $G_k(\F^n)\simeq G_{n-k}(\F^n)$ \end{zad} \begin{zad} Prove that $G_k(\F^n)$ is a connected, compact manifold of dimension $dk(n-k)$. \end{zad} \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 $G_k(\F^n)$. \end{zad} \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. \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. \end{zad} They are examples of coadjoint orbits \cite{Kirillov2004} \cite{Milnor&Stasheff1974} {{beginthm|Theorem}} {{endthm}} </wikitex> == Construction and examples == <wikitex>; ... </wikitex> == Invariants == <wikitex>; ... </wikitex> == Classification/Characterization == <wikitex>; ... </wikitex> == Further discussion == <wikitex>; ... </wikitex> == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]]<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> {{Stub}} == Introduction == <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. {{beginthm|Theorem|{{cite|}}}} $G_k(V)$ is a Hausdorff, compact space. \end{zad} \begin{zad} Prove that there exist a homeomorphism $G_k(\F^n)\simeq G_{n-k}(\F^n)$ \end{zad} \begin{zad} Prove that $G_k(\F^n)$ is a connected, compact manifold of dimension $dk(n-k)$. \end{zad} \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 $G_k(\F^n)$. \end{zad} \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. \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. \end{zad} They are examples of coadjoint orbits \cite{Kirillov2004} \cite{Milnor&Stasheff1974} {{beginthm|Theorem}} {{endthm}} </wikitex> == Construction and examples == <wikitex>; ... </wikitex> == Invariants == <wikitex>; ... </wikitex> == Classification/Characterization == <wikitex>; ... </wikitex> == Further discussion == <wikitex>; ... </wikitex> == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]]K$ be the real, complex or quaternion field and $V$ a vector space over $K$ of dimension $n$ and let

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$k\leq n$. A Grassmann manifolds of $k$-dimensional subspaces is a set

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$G_k(V)$ of $k$-dimensional subspaces. The set

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$G_k(V)$ is a quotient of a subset of

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$V\times ...\times V$ consisting of linearly independent $k$-tuples of vectors with the subspace topology. We define topology on

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$G_k(V)$ as the quotient topology. Note that the Grassmann manifold

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$G_k(V)$ around

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$W\in G_k(V)$ is locally modelled on the vector space

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$Hom (W^\bot ,W).$ </div> Grassmann manifold is a homogeneous space of the general linear group. General linear group

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$GL(V)$ acts transitively on

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$G_k(V)$ 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

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$O(V)$ acts transitively and the isotropy group of $W$ is $\O(W^\bot)\times O(W)$. The Grassmannians

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$G_1(V)$ are the well-known projective spaces, denoted $P(V)$. Note that

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$G_1(F^2)=S^d$ and if we identify

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$S^d$ with the one-point compactification of

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$\F$ the projection $p$ corresponds to the map

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$p_d :S^{2d-1}\to S^d$ given by

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$p_d(z_0,z_1)=z_0/z_1$ where

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$z_i\in\F$. Note, that the same formula works for

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$\F=\OO$, however the higher dimensional projective spaces over octonions do not exist. The maps

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$p_d :S^{2d-1}\to S^d$ for

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$d=1,2,4,8$ are called the Hopf maps and they play a very important role in homotopy theory; a fiber of

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$p_d$ is a sphere

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$S^{d-1}$. Check directly that the Hopf maps are locally trivial, thus fibrations.

There is an embedding of the Grassmannian

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$G_k(\F^n)$ in the cartesian space

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$\F^{n^2}=\Hom\,(F^n,F^n)$ which assigns to every subsapce the orthogonal projection on it. The embedding defines a natural (operator) metric on

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$G_k(\F^n)$.

Prove that there is a free action of the group

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$O(k,\F)$ on

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$V_k(\F^n)$ sucht that the orbit space is homeomorphic to

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$G_k(\F^n)$. Similarly for the noncompact Stiefel manifold.

\end{zad}

Prove that the map

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$p:V_k(\F^n)\to G_k(\F^n)$ is locally trivial (even a principal

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$O(k,\F)$-bundle), thus a fibration.

They are examples of coadjoint orbits [Kirillov2004]

[Milnor&Stasheff1974]

2 Construction and examples

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3 Invariants

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4 Classification/Characterization

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5 Further discussion

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6 References

• [Kirillov2004] A. A. Kirillov, Lectures on the orbit method, Graduate Studies in Mathematics 64, American Mathematical Society, Providence, RI, 2004. MR2069175 (2005c:22001) Zbl 02121486
• [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
• [[Template:{Milnor&Stasheff1974|[{Milnor&Stasheff1974]]] {{{Milnor&Stasheff1974}}