Grassmann manifolds
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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: $TG_k(V)= \Hom (\gamma_k^\bot , \gamma_k).$ | ||
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{{beginthm|Proposition|}} There exist a natural diffeomorphism $G_k(V)\simeq G_{n-k}(V^*)$. {{endthm}} | {{beginthm|Proposition|}} There exist a natural diffeomorphism $G_k(V)\simeq G_{n-k}(V^*)$. {{endthm}} | ||
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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 $G_k(V)$. | ||
− | 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 $G_k(\F{\infty} ).$ | + | 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 $G_k(\F{\infty} ).$ |
Revision as of 19:48, 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. They play a key role in topology and geometry as the universal spaces of vector bundles.
2 Construction and examples
Tex syntax errorof dimension and let
Tex syntax error. A Grassmann manifolds 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.
Theorem 2.1 [Milnor&Stasheff1974].
Tex syntax erroris a Hausdorff, compact space.
Theorem 2.2 [Milnor&Stasheff1974].
Note that the Grassmann manifold Tex syntax erroris a connected, compact 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 errorGrassmann manifold is a homogeneous space of the general linear group. General linear group
Tex syntax erroracts 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. The Grassmann manifold is equipped with the canonical, tautological vector 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:
Proposition 2.3. There exist a natural diffeomorphism
The Grassmannians Tex syntax error.
Tex syntax errorare projective spaces, denoted
Tex syntax error. Note that
Tex syntax error, where
Tex syntax error. 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
Tex syntax error. There is an embedding of the Grassmannian
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.
Infinite dimensional Grassmannians. Natural inclusions of vector space defines inclusions of Grassmannians. The colimit of the resulting sequence is denoted
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