Grassmann manifolds

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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.

2 Construction and examples

Let $\F=\Rr ,\Cc , \Hh$${{Stub}} == Introduction == <wikitex include="TeXInclude:Grassmann_manifolds">; 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. </wikitex> == Construction and examples == <wikitex include="TeXInclude:Grassmann_manifolds">; 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 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|Milnor&Stasheff1974}}}} $G_k(V)$ is a connected, compact smooth manifold of dimension $dk(n-k)$. For $\F=\Cc ,\Hh$ it is also a complex manifold.{{endthm}} 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).$ 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)$. 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 tangent bundle to Grassmaniann can be expressed in terms of the canonical bundle: $TG_k(V)= \Hom (\gamma_k^\bot , \gamma_k).$ {{beginthm|Proposition|}} There exist a natural diffeomorphism $G_k(V)\simeq G_{n-k}(V^*)$. {{endthm}} 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 $\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}$. 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)$. 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} ).$ == Invariants == <wikitex include="TeXInclude:Grassmann_manifolds">; Homotopy groups: Bot periodicity, relation to homotopy group of spheres Cohomology groups. ... </wikitex> == Classification/Characterization == <wikitex include="TeXInclude:Grassmann_manifolds">; ... </wikitex> == Further discussion == <wikitex>; Grassmann manifolds are examples of [[coadjoint orbits]] \cite{Kirillov2004}. </wikitex> == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]]{{Stub}} == Introduction == <wikitex include="TeXInclude:Grassmann_manifolds">; 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. </wikitex> == Construction and examples == <wikitex include="TeXInclude:Grassmann_manifolds">; 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 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|Milnor&Stasheff1974}}}} $G_k(V)$ is a connected, compact smooth manifold of dimension $dk(n-k)$. For $\F=\Cc ,\Hh$ it is also a complex manifold.{{endthm}} 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).$ 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)$. 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 tangent bundle to Grassmaniann can be expressed in terms of the canonical bundle: $TG_k(V)= \Hom (\gamma_k^\bot , \gamma_k).$ {{beginthm|Proposition|}} There exist a natural diffeomorphism $G_k(V)\simeq G_{n-k}(V^*)$. {{endthm}} 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 $\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}$. 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)$. 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} ).$ == Invariants == <wikitex include="TeXInclude:Grassmann_manifolds">; Homotopy groups: Bot periodicity, relation to homotopy group of spheres Cohomology groups. ... </wikitex> == Classification/Characterization == <wikitex include="TeXInclude:Grassmann_manifolds">; ... </wikitex> == Further discussion == <wikitex>; Grassmann manifolds are examples of [[coadjoint orbits]] \cite{Kirillov2004}. </wikitex> == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]]\F=\Rr ,\Cc , \Hh$ be the real, complex or quaternion field and $V$$V$ a vector space over $\F$$\F$ of dimension $n$$n$ and let $k\leq n$$k\leq n$. A Grassmann manifolds of $k$$k$-dimensional subspaces is a set $G_k(V)$$G_k(V)$ of $k$$k$-dimensional subspaces. The set $G_k(V)$$G_k(V)$ is a quotient of a subset of $V\times ...\times V$$V\times ...\times V$ consisting of linearly independent $k$$k$-tuples of vectors with the subspace topology. We define topology on $G_k(V)$$G_k(V)$ as the quotient topology.

Theorem 2.1 [Milnor&Stasheff1974]. $G_k(V)$$G_k(V)$ is a Hausdorff, compact space.

Theorem 2.2 [Milnor&Stasheff1974]. $G_k(V)$$G_k(V)$ is a connected, compact smooth manifold of dimension $dk(n-k)$$dk(n-k)$. For $\F=\Cc ,\Hh$$\F=\Cc ,\Hh$ it is also a complex manifold.

Note that the Grassmann manifold $G_k(V)$$G_k(V)$ around $W\in G_k(V)$$W\in G_k(V)$ is locally modelled on the vector space $Hom (W^\bot ,W).$$Hom (W^\bot ,W).$

Grassmann manifold is a homogeneous space of the general linear group. General linear group

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$GL(V)$ acts transitively on $G_k(V)$$G_k(V)$ with an isotropy group consisting of automorphisms preserving a given subspace. If the space $V$$V$ is equipped with a scalar product (hermitian metric resp.) then the group of isometries $O(V)$$O(V)$ acts transitively and the isotropy group of $W$$W$ is $O(W^\bot)\times O(W)$$O(W^\bot)\times O(W)$.

The Grassmann manifold is equipped with the canonical, tautological vector bundle $\gamma^V_k.$$\gamma^V_k.$ which is a subbundle of the trivial bundle $G_k(V)\times V \to G_k(V)\times V$$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 \}$$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 tangent bundle to Grassmaniann can be expressed in terms of the canonical bundle: $TG_k(V)= \Hom (\gamma_k^\bot , \gamma_k).$$TG_k(V)= \Hom (\gamma_k^\bot , \gamma_k).$

Proposition 2.3. There exist a natural diffeomorphism $G_k(V)\simeq G_{n-k}(V^*)$$G_k(V)\simeq G_{n-k}(V^*)$.

The Grassmannians $G_1(V)$$G_1(V)$ are projective spaces, denoted $P (V)$$P (V)$. Note that $G_1(F^2)=S^d$$G_1(F^2)=S^d$, where

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$d=dim_R F$. If we identify $S^d$$S^d$ with the one-point compactification of $\F$$\F$ the projection of the canonical principal bundle corresponds to the map $p_d :S^{2d-1}\to S^d$$p_d :S^{2d-1}\to S^d$ given by $p_d(z_0,z_1)=z_0/z_1$$p_d(z_0,z_1)=z_0/z_1$ where $z_i\in\F$$z_i\in\F$. Note, that the same formula works for octonions $\Oo$$\Oo$, however the higher dimensional projective spaces over octonions do not exist. The maps $p_d :S^{2d-1}\to S^d$$p_d :S^{2d-1}\to S^d$ for $d= 2,4,8$$d= 2,4,8$ are called the Hopf maps and they play a very important role in homotopy theory; a fibre of $p_d$$p_d$ is a sphere $S^{d-1}$$S^{d-1}$.

There is an embedding of the Grassmannian $G_k(V)$$G_k(V)$ in the Cartesian space $\F^{n^2}=\Hom\,(F^n,F^n)$$\F^{n^2}=\Hom\,(F^n,F^n)$ which assigns to every subspace the orthogonal projection on it. If $V$$V$ is equipped with a norm, the embedding defines a natural (operator) metric on $G_k(V)$$G_k(V)$.

Infinite dimensional Grassmannians. Natural inclusions of vector space $\F ^1 \subset \F ^2 \subset ...\F ^n \subset ...$$\F ^1 \subset \F ^2 \subset ...\F ^n \subset ...$ defines inclusions of Grassmannians. The colimit of the resulting sequence is denoted $G_k(\F{\infty} ).$$G_k(\F{\infty} ).$

Invariants

Homotopy groups: Bot periodicity, relation to homotopy group of spheres

Cohomology groups.

...

3 Classification/Characterization

...

4 Further discussion

Grassmann manifolds are examples of coadjoint orbits [Kirillov2004].

5 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