Grassmann manifolds
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== Introduction == | == Introduction == | ||
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| − | Grassmann manifolds | + | 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 $V$ a vector space over $K$. A Grassmann manifolds of $k$-dimensional subspaces is a set $G_k(V)$ of $k$-dimensional subspaces. The set $G_k(V)$ is a quotient of a subset of $V\times ...\times V$ consisting of linearly independent $k$-tuples of vectors with the subspace topology. We define topology on $G_k(V)$ as the quotient topology. |
| − | They are examples of | + | |
| + | {{beginthm|Theorem|{{cite|}}}} $G_k(V)$ is a Hausdorff, compact space. \end{zad} | ||
| + | |||
| + | \begin{zad} Prove that there exist a homeomorphism $G_k(\F^n)\simeq G_{n-k}(\F^n)$ \end{zad} | ||
| + | |||
| + | \begin{zad} Prove that $G_k(\F^n)$ is a connected, compact manifold of dimension $dk(n-k)$. \end{zad} | ||
| + | |||
| + | \begin{zad} There is an embedding of the Grassmannian $G_k(\F^n)$ in the cartesian space $\F^{n^2}=\Hom\,(F^n,F^n)$ which assigns to every subsapce the orthogonal projection on it. The embedding defines a natural (operator) metric on $G_k(\F^n)$. | ||
| + | \end{zad} | ||
| + | |||
| + | \begin{zad} The Grassmannians $G_1(\F^n)$ are the well-known projective spaces, denoted $\F P(n)$. Note that $G_1(\F^2)=S^d$ and if we identify $S^d$ with the one-point compactification of $\F$ the projection $p$ corresponds to the map $p_d :S^{2d-1}\to S^d$ given by $p_d(z_0,z_1)=z_0/z_1$ where $z_i\in\F$. Note, that the same formula works for $\F=\OO$, however the higher dimensional projective spaces over octonions do not exist. The maps $p_d :S^{2d-1}\to S^d$ for $d=1,2,4,8$ are called the Hopf maps and they play a very important role in homotopy theory; a fiber of $p_d$ is a sphere $S^{d-1}$. Check directly that the Hopf maps are locally trivial, thus fibrations. \end{zad} | ||
| + | |||
| + | \begin{zad} The natural action of $GL(n,\F)$ (resp. $O(n,\F)$) on $\F^n$ induces an action on $G_k(\F^n)$. 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 $O(k,\F)$ on $V_k(\F^n)$ sucht that the orbit space is homeomorphic to $G_k(\F^n)$. Similarly for the noncompact Stiefel manifold. | ||
| + | \end{zad} | ||
| + | |||
| + | \begin{zad} Prove that the map $p:V_k(\F^n)\to G_k(\F^n)$ is locally trivial (even a principal $O(k,\F)$-bundle), thus a fibration. | ||
| + | \end{zad} | ||
| + | |||
| + | They are examples of coadjoint orbits \cite{Kirillov2004} | ||
\cite{Milnor&Stasheff1974} | \cite{Milnor&Stasheff1974} | ||
Revision as of 12:59, 26 November 2010
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This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
1 Introduction
be the real, complex or quaternion field and 
a vector space over . 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 1.1 [[[#|]]]. 
corresponds to the map 
)
Tex syntax erroris 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 erroris a connected, compact manifold of dimension
Tex syntax error. \end{zad} \begin{zad} There is an embedding of the Grassmannian
Tex syntax errorin the cartesian space
Tex syntax errorwhich 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 GrassmanniansTex syntax errorare the well-known projective spaces, denoted
Tex syntax error. Note that
Tex syntax errorand if we identify
Tex syntax errorwith the one-point compactification of
Tex syntax errorthe projection
corresponds to the map Tex syntax errorgiven by
Tex syntax errorwhere
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 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. 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 errorinduces an action on
Tex syntax error. Show that the actions are transitive and describe the isotropy groups (in particular of the canonical subspace
)
\end{zad}
\begin{zad} Prove that there is a free action of the groupTex syntax erroron
Tex syntax errorsucht that the orbit space is homeomorphic to
Tex syntax error. Similarly for the noncompact Stiefel manifold.
\end{zad}
\begin{zad} Prove that the mapTex syntax erroris locally trivial (even a principal
Tex syntax error-bundle), thus a fibration.
\end{zad}
They are examples of coadjoint orbits [Kirillov2004]
Theorem 1.2.
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
