Grassmann manifolds

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{{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}}
{{beginthm|Theorem|{{cite|Milnor&Stasheff1974}}}} $G_k(V)$ is a connected, compact smooth manifold of dimension $dk(n-k)$. For $\F=\C,\H$ it is also a complex manifold.{{endthm}}
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{{beginthm|Theorem|{{cite|Milnor&Stasheff1974}}}} $G_k(V)$ is a connected, compact smooth manifold of dimension $dk(n-k)$. For $\F=\Cc,\Hh$ it is also a complex manifold.{{endthm}}
Note that the Grassmann manifold $G_k(V)$ around $W\in G_k(V)$ is locally modelled on the vector space $Hom (W^\bot ,W).$ {{endthm}}
Note that the Grassmann manifold $G_k(V)$ around $W\in G_k(V)$ is locally modelled on the vector space $Hom (W^\bot ,W).$ {{endthm}}

Revision as of 18:03, 26 November 2010

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
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be the real, complex or quaternion field and
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a vector space over
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of dimension
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and let
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. A Grassmann manifolds of
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-dimensional subspaces is a set
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of
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-dimensional subspaces. The set
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is a quotient of a subset of
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consisting of linearly independent
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-tuples of vectors with the subspace topology. We define topology on
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as the quotient topology.
Theorem 1.1 [Milnor&Stasheff1974].
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is a Hausdorff, compact space.
Theorem 1.2 [Milnor&Stasheff1974].
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is a connected, compact smooth manifold of dimension
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. For
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it is also a complex manifold.
Note that the Grassmann manifold
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around
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is locally modelled on the vector space
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</div> Grassmann manifold is a homogeneous space of the general linear group. General linear group
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acts transitively on
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with an isotropy group consisting of automorphisms preserving a given subspace. If the space
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is equipped with a scalar product (hermitian metric resp.) then the group of isometries
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acts transitively and the isotropy group of
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is
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. The Grassmann manifold is equipped with the canonical, tautological vector bundle
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which is a subbundle of the trivial bundle
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. The total space is
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The total space of the associated principal bundle is a Stiefel manifold.
Proposition 1.3. There exist a natural diffeomorphism
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.
The Grassmannians
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are projective spaces, denoted
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. Note that
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, where
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. If we identify
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with the one-point compactification of
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the projection of the canonical principal bundle corresponds to the map
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given by
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where
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. Note, that the same formula works for octonions, however the higher dimensional projective spaces over octonions do not exist. The maps
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for
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are called the Hopf maps and they play a very important role in homotopy theory; a fiber of
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is a sphere
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. There is an embedding of the Grassmannian
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in the cartesian space
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which assigns to every subspace the orthogonal projection on it. If
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is equipped with a norm, 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.

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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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