Grassmann manifolds

From Manifold Atlas
Revision as of 11:59, 26 November 2010 by Stefan Jackowski (Talk | contribs)
Jump to: navigation, search


This page has not been refereed. The information given here might be incomplete or provisional.

Contents

1 Introduction

Grassmann manifolds are named after Hermann Grassmann, a German school teacher in Stettin who developed basic notions of linear algebra. Let K be the real, complex or quaternion field and
Tex syntax error
a vector space over K. A Grassmann manifolds of
Tex syntax error
-dimensional subspaces is a set
Tex syntax error
of
Tex syntax error
-dimensional subspaces. The set
Tex syntax error
is a quotient of a subset of
Tex syntax error
consisting of linearly independent
Tex syntax error
-tuples of vectors with the subspace topology. We define topology on
Tex syntax error
as the quotient topology.


Theorem 1.1 [[[#|]]].
Tex syntax error
is a Hausdorff, compact space. \end{zad} \begin{zad} Prove that there exist a homeomorphism
Tex syntax error
\end{zad} \begin{zad} Prove that
Tex syntax error
is a connected, compact manifold of dimension
Tex syntax error
. \end{zad} \begin{zad} There is an embedding of the Grassmannian
Tex syntax error
in the cartesian space
Tex syntax error
which assigns to every subsapce the orthogonal projection on it. The embedding defines a natural (operator) metric on
Tex syntax error
.

\end{zad}

\begin{zad} The Grassmannians
Tex syntax error
are the well-known projective spaces, denoted
Tex syntax error
. Note that
Tex syntax error
and if we identify
Tex syntax error
with the one-point compactification of
Tex syntax error
the projection p corresponds to the map
Tex syntax error
given by
Tex syntax error
where
Tex syntax error
. Note, that the same formula works for
Tex syntax error
, however the higher dimensional projective spaces over octonions do not exist. The maps
Tex syntax error
for
Tex syntax error
are called the Hopf maps and they play a very important role in homotopy theory; a fiber of
Tex syntax error
is a sphere
Tex syntax error
. Check directly that the Hopf maps are locally trivial, thus fibrations. \end{zad} \begin{zad} The natural action of
Tex syntax error
(resp.
Tex syntax error
) on
Tex syntax error
induces an action on
Tex syntax error
. Show that the actions are transitive and describe the isotropy groups (in particular of the canonical subspace F^k\subset F^n)

\end{zad}

\begin{zad} Prove that there is a free action of the group
Tex syntax error
on
Tex syntax error
sucht that the orbit space is homeomorphic to
Tex syntax error
. Similarly for the noncompact Stiefel manifold.

\end{zad}

\begin{zad} Prove that the map
Tex syntax error
is locally trivial (even a principal
Tex syntax error
-bundle), thus a fibration.

\end{zad}

They are examples of coadjoint orbits [Kirillov2004]

[Milnor&Stasheff1974]

Theorem 1.2.



2 Construction and examples

...

3 Invariants

...

4 Classification/Characterization

...

5 Further discussion

...

6 References

Personal tools
Namespaces
Variants
Actions
Navigation
Interaction
Toolbox