Grassmann manifolds
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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.
![G_k(V)](/images/math/b/2/9/b29c11c1c9a37cae146d723a772dccd8.png)
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
.
![G_k(V)\simeq G_{n-k}(V^*)](/images/math/7/f/0/7f0c2f05ace8b18ba6ae661b706d7d28.png)
![G_1(V)](/images/math/a/5/8/a5839f34b3976fe4a7f1317bf772b36f.png)
![P(V)](/images/math/f/a/e/fae268d5078da3d832516eb0714cbc3a.png)
![G_1(F^2)=S^d](/images/math/e/0/f/e0f878de2d9a04ef68a5447a7837794d.png)
![S^d](/images/math/9/e/a/9eaa915074ddeb6ea9dd3fac0e6927ef.png)
![\F](/images/math/f/a/c/facab4c79138bc48b7c0b9c5642f2f5c.png)
![p](/images/math/2/a/0/2a039ed8fdbf4ceaa9e79cdc3aecd1a2.png)
![p_d :S^{2d-1}\to S^d](/images/math/6/8/b/68b56784bc99ad8c36e322cf25133980.png)
![p_d(z_0,z_1)=z_0/z_1](/images/math/c/8/0/c80fe06240a70601b13135373c3a4238.png)
![z_i\in\F](/images/math/9/c/c/9cc9b180553434117d88d97a308a7aa6.png)
Tex syntax error, however the higher dimensional projective spaces over octonions do not exist. The maps
![p_d :S^{2d-1}\to S^d](/images/math/6/8/b/68b56784bc99ad8c36e322cf25133980.png)
![d=1,2,4,8](/images/math/1/3/7/137c369122e4c367c03a8c6b66a49024.png)
![p_d](/images/math/8/6/1/861e562c9dc1b636d0e84f92ee37f4b3.png)
![S^{d-1}](/images/math/a/f/e/afeeaed682327313e2158c9c9e27bdb7.png)
There is an embedding of the Grassmannianin the cartesian space
which assigns to every subsapce the orthogonal projection on it. The embedding defines a natural (operator) metric on
.
Prove that there is a free action of the group on
sucht that the orbit space is homeomorphic to
. Similarly for the noncompact Stiefel manifold.
\end{zad}
Prove that the map is locally trivial (even a principal
-bundle), thus a fibration.
They are examples of coadjoint orbits [Kirillov2004]
Theorem 1.4.
2 Construction and examples
...
3 Invariants
...
4 Classification/Characterization
...
5 Further discussion
...
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
- [[Template:{Milnor&Stasheff1974|[{Milnor&Stasheff1974]]] {{{Milnor&Stasheff1974}}