Grassmann manifolds
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{{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}} | ||
− | {{beginthm|Theorem|{{cite|Milnor&Stasheff1974}}}} $G_k(V)$ is a connected, compact smooth manifold of dimension $dk(n-k)$. For $\F=\ | + | {{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 $Hom (W^\bot ,W).$ {{endthm}} | Note that the Grassmann manifold $G_k(V)$ around $W\in G_k(V)$ is locally modelled on the vector space $Hom (W^\bot ,W).$ {{endthm}} |
Revision as of 19:03, 26 November 2010
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Contents |
1 Introduction
Tex syntax errorbe the real, complex or quaternion field and
Tex syntax errora vector space over
Tex syntax errorof dimension
Tex syntax errorand let
Tex syntax error. A Grassmann manifolds of
Tex syntax error-dimensional subspaces is a set
Tex syntax errorof
Tex syntax error-dimensional subspaces. The set
Tex syntax erroris a quotient of a subset of
Tex syntax errorconsisting of linearly independent
Tex syntax error-tuples of vectors with the subspace topology. We define topology on
Tex syntax erroras the quotient topology.
Theorem 1.1 [Milnor&Stasheff1974].
Tex syntax erroris a Hausdorff, compact space.
Theorem 1.2 [Milnor&Stasheff1974].
Note that the Grassmann manifold Tex syntax erroris a connected, compact smooth manifold of dimension
Tex syntax error. For
Tex syntax errorit is also a complex manifold.
Tex syntax erroraround
Tex syntax erroris locally modelled on the vector space
Tex syntax error</div> Grassmann 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
Tex syntax erroris equipped with a scalar product (hermitian metric resp.) then the group of isometries
Tex syntax erroracts transitively and the isotropy group of
Tex syntax erroris
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.
Proposition 1.3. 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
Tex syntax errorare called the Hopf maps and they play a very important role in homotopy theory; a fiber 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
Tex syntax erroris equipped with a norm, the embedding defines a natural (operator) metric on
Tex syntax error. Prove that there is a free action of the group
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