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. | + | 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. |
{{beginthm|Theorem|{{cite|Milnor&Stasheff1974}}}} $G_k(V)$ is a Hausdorff, compact space. {{endthm}} | {{beginthm|Theorem|{{cite|Milnor&Stasheff1974}}}} $G_k(V)$ is a Hausdorff, compact space. {{endthm}} | ||
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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| | + | {{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, 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 $S^{d-1}$. | 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 $S^{d-1}$. | ||
− | There is an embedding of the Grassmannian $G_k( | + | 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 $G_k(V)$. |
Revision as of 18:48, 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 be the real, complex or quaternion field and a vector space over of dimension and let . A Grassmann manifolds of -dimensional subspaces is a set of -dimensional subspaces. The set is a quotient of a subset of consisting of linearly independent -tuples of vectors with the subspace topology. We define topology on as the quotient topology.
Note that the Grassmann manifold around is locally modelled on the vector space </div>
Grassmann manifold is a homogeneous space of the general linear group. General linear groupTex syntax erroracts transitively on with 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 acts transitively and the isotropy group of is .
The Grassmann manifold is equipped with the canonical, tautological vector bundle which is a subbundle of the trivial bundle . The total space is The total space of the associated principal bundle is a Stiefel manifold.
Tex syntax error. If we identify with the one-point compactification of the projection of the canonical principal bundle corresponds to the map given by where . Note, that the same formula works for octonions, however the higher dimensional projective spaces over octonions do not exist. The maps for
Tex syntax errorare called the Hopf maps and they play a very important role in homotopy theory; a fiber of is a sphere .
There is an embedding of the Grassmannian in the cartesian space which 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 erroron
Tex syntax errorsucht that the orbit space is homeomorphic to
Tex syntax error. Similarly for the noncompact Stiefel manifold.
\end{zad}
Prove that the mapTex syntax erroris locally trivial (even a principal
Tex syntax error-bundle), thus a fibration.
2 Construction and examples
...
3 Invariants
...
4 Classification/Characterization
...
5 Further discussion
Grassmann manifolds are examples of coadjoint orbits [Kirillov2004].
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