Grassmann manifolds
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== Introduction == | == Introduction == | ||
<wikitex include="TeXInclude:Grassmann_manifolds">; | <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. See also [[Wikipedia:Grassmannian|Grassmannian]] | |
− | 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. | + | |
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</wikitex> | </wikitex> | ||
== Construction and examples == | == Construction and examples == | ||
<wikitex include="TeXInclude:Grassmann_manifolds">; | <wikitex include="TeXInclude:Grassmann_manifolds">; | ||
+ | </wikitex> | ||
− | 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 | + | === Construction === |
− | + | <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 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 $O(W^\bot)\times O(W)$. | |
− | {{beginthm|Theorem|{{cite|Milnor&Stasheff1974}}}} $G_k(V)$ is a | + | {{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}} |
− | Note that the Grassmann manifold $G_k(V)$ around $W\in G_k(V)$ is locally modelled on the vector space $Hom (W^\bot ,W).$ | + | 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}} | ||
+ | </wikitex> | ||
+ | === The canonical bundle === | ||
+ | <wikitex include="TeXInclude:Grassmann_manifolds">; | ||
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. | ||
− | The tangent bundle to Grassmaniann can be expressed in terms of the canonical bundle: $TG_k(V)= \Hom (\gamma_k^\bot , \gamma_k).$ | + | The tangent bundle to Grassmaniann can be expressed in terms of the canonical bundle: $TG_k(V)= \Hom (\gamma_k^\bot , \gamma_k).$</wikitex> |
− | + | === Low dimensional Grassmannians === | |
− | + | <wikitex include="TeXInclude:Grassmann_manifolds">; | |
− | + | 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 $S^{d-1}$. | |
− | The Grassmannians $G_1(V)$ are projective spaces, denoted $P (V)$. Note that $G_1(F^2)=S^d$, where $d= | + | </wikitex> |
− | + | === Embeddings of Grassmannians into affine and projective space === | |
+ | <wikitex include="TeXInclude:Grassmann_manifolds">; | ||
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 $G_k(V)$. | 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 $G_k(V)$. | ||
− | + | </wikitex> | |
− | 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 $B\GL (\F).$ | + | === Infinite dimensional Grassmannians === |
− | + | <wikitex include="TeXInclude:Grassmann_manifolds">; | |
+ | 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 $B\GL (\F).$ | ||
+ | </wikitex> | ||
== Invariants == | == Invariants == | ||
+ | <wikitex include="TeXInclude:Grassmann_manifolds">; | ||
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+ | === Homotopy groups === | ||
<wikitex include="TeXInclude:Grassmann_manifolds">; | <wikitex include="TeXInclude:Grassmann_manifolds">; | ||
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. | 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. | ||
− | {{beginthm|Proposition|{(R.Bott)}}} For each $i>0$ there are isomorphisms: $\pi_i(BGL(\Rr) \simeq \pi_{i+8}(BGL(\Rr)$ and $\pi_i(BGL(\Cc) \simeq \pi_{i+2}(BGL(\Cc)$ | + | {{beginthm|Proposition|{(R.Bott)}}} For each $i>0$ there are isomorphisms: $\pi_i(BGL(\Rr)) \simeq \pi_{i+8}(BGL(\Rr))$ and $\pi_i(BGL(\Cc)) \simeq \pi_{i+2}(BGL(\Cc))$ |
{{endthm}} | {{endthm}} | ||
+ | </wikitex> | ||
− | Cohomology groups | + | === Cohomology groups === |
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== Further discussion == | == Further discussion == | ||
− | <wikitex>; | + | <wikitex include="TeXInclude:Grassmann_manifolds">; |
Grassmann manifolds are examples of [[coadjoint orbits]] \cite{Kirillov2004}. | Grassmann manifolds are examples of [[coadjoint orbits]] \cite{Kirillov2004}. | ||
</wikitex> | </wikitex> |
Latest revision as of 19:05, 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
Tex syntax errorof dimension and let . A Grassmannian of -dimensional subspaces is a set
Tex syntax errorof -dimensional subspaces. The set
Tex syntax erroris a quotient of a subset of
Tex syntax errorconsisting of linearly independent -tuples of vectors with the subspace topology. We define topology on
Tex syntax erroras the quotient topology. Grassmannian is a homogeneous space of the general linear group. General linear group acts transitively on
Tex syntax errorwith an isotropy group consisting of automorphisms preserving a given subspace. If the space is equipped with a scalar product (hermitian metric resp.) then the group of isometries
Tex syntax erroracts transitively and the isotropy group of is
Tex syntax error.
Tex syntax erroris a Hausdorff, compact, connected smooth manifold of dimension
Tex syntax error. For it is also a complex manifold.
Tex syntax erroraround
Tex syntax erroris locally modelled on the vector space
Tex syntax error
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[edit] 2.2 The canonical bundle
Tex syntax errorwhich is a subbundle of the trivial bundle
Tex syntax error. The total space is
Tex syntax errorThe 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:
[edit] 2.3 Low dimensional Grassmannians
Tex syntax errorare projective spaces, denoted
Tex syntax error. Note that
Tex syntax error, where . If we identify
Tex syntax errorwith the one-point compactification of
Tex syntax errorthe projection of the canonical principal bundle corresponds to the map
Tex syntax errorgiven by
Tex syntax errorwhere
Tex syntax error. Note, that the same formula works for octonions , however the higher dimensional projective spaces over octonions do not exist. The maps
Tex syntax errorfor are called the Hopf maps and they play a very important role in homotopy theory; a fibre of
Tex syntax erroris a sphere .
[edit] 2.4 Embeddings of Grassmannians into affine and projective space
Tex syntax errorin the Cartesian space
Tex syntax errorwhich assigns to every subspace the orthogonal projection on it. If 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 defines inclusions of Grassmannians. The colimit of the resulting sequence is denoted and also . One can also take the colimit with respect to both dimension of the space and of the subspaces. We have a sequence of inclusions and its colimit is denoted
[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 do not depend on , if Homotopy groups in the stable range are described by the Bott periodicity theorem.
Proposition 3.1 {(R.Bott).} For each there are isomorphisms: and
[edit] 3.2 Cohomology groups
...
[edit] 4 Classification/Characterization
...
[edit] 5 Further discussion
Grassmann manifolds are examples of coadjoint orbits [Kirillov2004].
[edit] 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
[edit] 7 External links
- The Wikipedia page on Grassmannian