Grassmann manifolds

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Contents

[edit] 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

[edit] 2 Construction and examples

[edit] 2.1 Construction

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 Grassmannian 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. Grassmannian is a homogeneous space of the general linear group. General linear group \GL(V) 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).

Theorem 2.1 [Milnor&Stasheff1974]. G_k(V) is a Hausdorff, compact, connected 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).

Proposition 2.2. There exist a natural diffeomorphism G_k(V)\simeq G_{n-k}(V^*).

[edit] 2.2 The canonical bundle

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).

[edit] 2.3 Low dimensional Grassmannians

The Grassmannians G_1(V) are projective spaces, denoted P (V). Note that G_1(F^2)=S^d, where d=dim_{\Rr} F. 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}.

[edit] 2.4 Embeddings of Grassmannians into affine and projective space

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).

[edit] 2.5 Infinite dimensional Grassmannians

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).

[edit] 3 Invariants


[edit] 3.1 Homotopy groups

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 3.1 {(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))

[edit] 3.2 Cohomology groups

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[edit] 4 Classification/Characterization

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[edit] 5 Further discussion

Grassmann manifolds are examples of coadjoint orbits [Kirillov2004].

[edit] 6 References

[edit] 7 External links

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