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 $ | + | 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}} |
Revision as of 18:29, 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 group acts 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.
The Grassmannians are projective spaces, denoted . Note that , where . 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 are 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 GrassmannianTex syntax errorin the cartesian space which assigns to every subsapce the orthogonal projection on it. The embedding defines a natural (operator) metric on
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Tex syntax erroron
Tex syntax errorsucht that the orbit space is homeomorphic to
Tex syntax error. Similarly for the noncompact Stiefel manifold.
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Prove that the mapTex syntax erroris locally trivial (even a principal
Tex syntax error-bundle), thus a fibration.
2 Construction and examples
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3 Invariants
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4 Classification/Characterization
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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