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.
2 Construction and examples
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)
![G_k(V)](/images/math/b/2/9/b29c11c1c9a37cae146d723a772dccd8.png)
![dk(n-k)](/images/math/9/1/1/9117cf1a1bf85fc3b3a7da1ed7e6a7d6.png)
![\F=\Cc ,\Hh](/images/math/3/5/9/359b4afae6830655f77de3e0a27fe7b8.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
.
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.
![G_k(V)\simeq G_{n-k}(V^*)](/images/math/7/f/0/7f0c2f05ace8b18ba6ae661b706d7d28.png)
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 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
.
Prove that there is a free action of the group on
such 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.
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