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. They play a key role in topology and geometry as the universal spaces of vector bundles. See also Grassmannian


2 Construction and examples

Let \F=\Rr ,\Cc , \Hh 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 2.1 [Milnor&Stasheff1974]. G_k(V) is a Hausdorff, compact space.
Theorem 2.2 [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.

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).

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.

The tangent bundle to Grassmaniann can be expressed in terms of the canonical bundle: TG_k(V)= \Hom (\gamma_k^\bot , \gamma_k).

Proposition 2.3. 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 \Oo, however the higher dimensional projective spaces over octonions do not exist. The maps p_d :S^{2d-1}\to S^d for d= 2,4,8 are called the Hopf maps and they play a very important role in homotopy theory; a fibre of p_d is a sphere S^{d-1}.

There is an embedding of the Grassmannian G_k(V) in the Cartesian space \F^{n^2}=\Hom\,(F^n,F^n) which assigns to every subspace the orthogonal projection on it. If V is equipped with a norm, the embedding defines a natural (operator) metric on G_k(V).

Infinite dimensional Grassmannians. Natural inclusions of vector space \F ^1 \subset \F ^2 \subset ...\F ^n \subset ... defines inclusions of Grassmannians. The colimit of the resulting sequence is denoted G_k(\F^{\infty} ) and also BGL(k,\F). One can also take the colimit with respect to both dimension of the space and of the subspaces. We have a sequence of inclusions G_1(\F^2)\subset G_2(F^4)\subset ... \subset G_n(F^2n) \subset ... and its colimit is denoted B\GL (\F).


Invariants

Homotopy groups of Grassmannians are closely related to homotopy groups of spheres via the appropriate fibration sequences. They also imply that the groups \pi_i(G_k (V)) do not depend on V, if k\leq\leq dim V. Homotopy groups in the stable range are described by the Bott periodicity theorem.

Proposition 4.1 {(R.Bott).} For each i>0 there are isomorphisms: \pi_i(BGL(\Rr) \simeq \pi_{i+8}(BGL(\Rr) and \pi_i(BGL(\Cc) \simeq \pi_{i+2}(BGL(\Cc)

Cohomology groups.

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3 Classification/Characterization

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4 Further discussion

Grassmann manifolds are examples of coadjoint orbits [Kirillov2004].

5 References

6 External links

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