Grassmann manifolds

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== Introduction ==
== Introduction ==
<wikitex>;
<wikitex>;
Grassmann manifolds are named after Hermann Grassmann, a German school teacher in Stettin who developed basic notions of linear algebra. Let $\F$ be the real, complex or quaternion field and $V$ a vector space over $K$ of dimension $n$ and let $k\leq n$. 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.
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Grassmann manifolds are named after Hermann Grassmann, a German school teacher in Stettin who developed basic notions of linear algebra. Let $\F$ be the real, complex or quaternion field and $V$ a vector space over $F$ of dimension $n$ and let $k\leq n$. 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.
{{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}}

Revision as of 17:29, 26 November 2010

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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 \F be the real, complex or quaternion field and V a vector space over F of dimension n and let k\leq n. 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.

Theorem 1.1 [Milnor&Stasheff1974]. G_k(V) is a Hausdorff, compact space.
Theorem 1.2 [Milnor&Stasheff1974]. G_k(V) is a connected, compact smooth manifold of dimension dk(n-k).

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). </div>

Grassmann manifold is a homogeneous space of the general linear group. General linear group
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acts transitively on G_k(V) with an isotropy group consisting of automorphisms preserving a given subspace. If the space V is equipped with a scalar product (hermitian metric resp.) then the group of isometries O(V) acts transitively and the isotropy group of W is O(W^\bot)\times O(W).

The Grassmann manifold is equipped with the canonical, tautological vector bundle \gamma^V_k. which is a subbundle of the trivial bundle G_k(V)\times V \to G_k(V)\times V. The total space is E(\gamma^V_k) = \{(W,w)\in G_k(V)\times V\, |\,\, w\in W \} The total space of the associated principal bundle is a Stiefel manifold.

Proposition 1.3 [Milnor&Stasheff1974]. There exist a natural diffeomorphism G_k(V)\simeq G_{n-k}(V^*).
The Grassmannians G_1(V) are projective spaces, denoted P (V). Note that G_1(F^2)=S^d, where
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. If we identify S^d with the one-point compactification of \F the projection of the canonical principal bundle 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 octonions, however the higher dimensional projective spaces over octonions do not exist. The maps p_d :S^{2d-1}\to S^d for
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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}. There is an embedding of the Grassmannian
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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
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.


Prove that there is a free action of the group
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on
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sucht that the orbit space is homeomorphic to
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. Similarly for the noncompact Stiefel manifold.

\end{zad}

Prove that the map
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is locally trivial (even a principal
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-bundle), thus a fibration.

2 Construction and examples

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3 Invariants

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4 Classification/Characterization

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5 Further discussion

Grassmann manifolds are examples of coadjoint orbits [Kirillov2004].

6 References

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