# Grassmann manifolds

## 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

### 2.1 Construction


Theorem 2.1 [Milnor&Stasheff1974]. $G_k(V)$$G_k(V)$ is a Hausdorff, compact, connected smooth manifold of dimension $dk(n-k)$$dk(n-k)$. For $\F=\Cc ,\Hh$$\F=\Cc ,\Hh$ it is also a complex manifold.

Note that the Grassmann manifold $G_k(V)$$G_k(V)$ around $W\in G_k(V)$$W\in G_k(V)$ is locally modelled on the vector space $Hom (W^\bot ,W).$$Hom (W^\bot ,W).$

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

### 2.2 The canonical bundle

The Grassmann manifold is equipped with the canonical, tautological vector bundle $\gamma^V_k.$$\gamma^V_k.$ which is a subbundle of the trivial bundle $G_k(V)\times V \to G_k(V)\times V$$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 \}$$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).$$TG_k(V)= \Hom (\gamma_k^\bot , \gamma_k).$

### 2.3 Low dimensional Grassmannians

The Grassmannians $G_1(V)$$G_1(V)$ are projective spaces, denoted $P (V)$$P (V)$. Note that $G_1(F^2)=S^d$$G_1(F^2)=S^d$, where $d=dim_{\Rr} F$$d=dim_{\Rr} F$. If we identify $S^d$$S^d$ with the one-point compactification of $\F$$\F$ the projection of the canonical principal bundle corresponds to the map $p_d :S^{2d-1}\to S^d$$p_d :S^{2d-1}\to S^d$ given by $p_d(z_0,z_1)=z_0/z_1$$p_d(z_0,z_1)=z_0/z_1$ where $z_i\in\F$$z_i\in\F$. Note, that the same formula works for octonions $\Oo$$\Oo$, however the higher dimensional projective spaces over octonions do not exist. The maps $p_d :S^{2d-1}\to S^d$$p_d :S^{2d-1}\to S^d$ for $d= 2,4,8$$d= 2,4,8$ are called the Hopf maps and they play a very important role in homotopy theory; a fibre of $p_d$$p_d$ is a sphere $S^{d-1}$$S^{d-1}$.

### 2.4 Embeddings of Grassmannians into affine and projective space

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

### 2.5 Infinite dimensional Grassmannians

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

## 3 Invariants

### 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))$$\pi_i(G_k (V))$ do not depend on $V$$V$, if $k\leq\leq dim V.$$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$$i>0$ there are isomorphisms: $\pi_i(BGL(\Rr)) \simeq \pi_{i+8}(BGL(\Rr))$$\pi_i(BGL(\Rr)) \simeq \pi_{i+8}(BGL(\Rr))$ and $\pi_i(BGL(\Cc)) \simeq \pi_{i+2}(BGL(\Cc))$$\pi_i(BGL(\Cc)) \simeq \pi_{i+2}(BGL(\Cc))$

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## 5 Further discussion

Grassmann manifolds are examples of coadjoint orbits [Kirillov2004].