Graßmann manifold

Graßmann manifolds (also Grassmann manifolds ) are a fundamental term in mathematics for both differential geometry and algebraic geometry . They parameterize the subspaces of a vector space and thus represent a generalization of the projective space . They are named after Hermann Graßmann .

definition

Let be a vector space over a body . Then designated ${\displaystyle V}$ ${\displaystyle \mathbb {K} }$

${\displaystyle Gr(r,V)}$

is the set of -dimensional subspaces of . If -dimensional, one also denotes with ${\displaystyle r}$ ${\displaystyle V}$ ${\displaystyle V}$ ${\displaystyle n}$ ${\displaystyle Gr(r,V)}$

${\displaystyle Gr(r,n)}$.

Effect of the orthogonal / unitary and linear group

In the case, the orthogonal group is effective ${\displaystyle \mathbb {K} =\mathbb {R} }$

${\displaystyle O(n)}$

on through ${\displaystyle Gr(r,n)}$

${\displaystyle (A,W)\rightarrow A(W)}$.

The effect is transitive , the stabilizers are conjugated to

${\displaystyle O(r)\times O(n-r)}$.

So you get a bijection between and the homogeneous space ${\displaystyle Gr(r,n)}$

${\displaystyle O(n)/O(r)\times O(n-r)=GL(n,\mathbb {R} )/GL(r,\mathbb {R} )\times GL(n-r,\mathbb {R} )}$.

In the case, the unitary group acts transitive and provides a bijection of the Graßmann manifold ${\displaystyle \mathbb {K} =\mathbb {C} }$ ${\displaystyle U(n)}$

${\displaystyle U(n)/U(r)\times U(n-r)=GL(n,\mathbb {C} )/GL(r,\mathbb {C} )\times GL(n-r,\mathbb {C} )}$.

Similarly, a bijection between and is obtained for any body ${\displaystyle \mathbb {K} }$ ${\displaystyle Gr(r,n)}$

${\displaystyle GL(n,\mathbb {K} )/GL(r,\mathbb {K} )\times GL(n-r,\mathbb {K} )}$.

topology

As a real Graßmann manifold (of the -dimensional subspaces in ) one designates with by the identification with ${\displaystyle r}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle Gr(r,n)}$

${\displaystyle O(n)/O(r)\times O(n-r)}$

given topology .

One calls a complex Graßmann manifold accordingly ${\displaystyle Gr(r,n)}$

${\displaystyle U(n)/U(r)\times U(n-r)}$.

Canonical inclusion induces inclusion . One defines ${\displaystyle \mathbb {K} ^{n}\subset \mathbb {K} ^{n+1}}$ ${\displaystyle Gr(r,n)\subset Gr(r,n+1)}$

${\displaystyle Gr(r,\infty ):=\lim _{n}Gr(r,n)}$

as inductive Limes the one with the Limes topology . ${\displaystyle Gr(r,n)}$

Algebraic variety

Grassmann manifolds are projective varieties using Plücker embedding .

Tautological bundle

Be the projective limit with respect to the canonical inclusions and define ${\displaystyle \mathbb {K} ^{\infty }:=\lim _{n}\mathbb {K} ^{n}}$

${\displaystyle \gamma ^{r}:=\left\{(W,x)\in Gr(r,\infty )\times \mathbb {K} ^{\infty }:x\in W\right\}\subset Gr(r,\infty )\times \mathbb {K} ^{\infty }}$.

Then the projection onto the first factor is a vector bundle

${\displaystyle \gamma ^{r}\rightarrow Gr(r,\infty )}$,

which is called a tautological or universal r-dimensional vector bundle .

Classifying figure

For every r-dimensional vector bundle there is a continuous mapping ${\displaystyle E\rightarrow B}$

${\displaystyle f\colon B\rightarrow Gr(r,\infty )}$,

so that the pullback of the tautological bundle is under . ${\displaystyle E}$ ${\displaystyle \gamma ^{r}}$ ${\displaystyle f}$

In the case of the tangent bundle of a differentiable manifold , one has the following explicit description of the classifying mapping: According to Whitney's embedding theorem , one can assume that a submanifold is one . The tangent plane at a point is then of the form ${\displaystyle TM}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle \mathbb {R} ^{m}}$ ${\displaystyle T_{x}M}$ ${\displaystyle x\in M}$

${\displaystyle T_{x}M=x+W_{x}}$

for a subspace . The assignment ${\displaystyle W_{x}\subset \mathbb {R} ^{m}}$

${\displaystyle x\rightarrow W_{x}}$

defines a continuous mapping

${\displaystyle f\colon M\rightarrow Gr(r,m)\subset Gr(r,\infty )}$

and you can show that

${\displaystyle f^{*}\gamma ^{r}=TM}$

is.

Classifying space for principal bundles

The Graßmann manifold is the classifying space for principal bundles with structural groups . And thus also for principal bundles with a structure group , because because the inclusion is a homotopy equivalence, each bundle can be reduced to the structure group . The following applies: ${\displaystyle Gr(r,\infty )}$ ${\displaystyle O(r)}$ ${\displaystyle GL(r)}$ ${\displaystyle O(r)\rightarrow GL(r)}$ ${\displaystyle GL(r)}$ ${\displaystyle O(r)}$

${\displaystyle Gr(r,\infty )\simeq BGL(r,\mathbb {K} )\simeq BO(r,\mathbb {K} )}$.

The canonical projection of the Stiefel manifold to which Repere maps on each of the generated by them subspace is the universal -bundle. (The tautological bundle results from the universal bundle as an associated vector bundle through the canonical effect of on the vector space .) ${\displaystyle V(r,\infty )}$ ${\displaystyle G(r,\infty )}$ ${\displaystyle O(r)}$ ${\displaystyle \gamma ^{r}}$ ${\displaystyle O(r)}$ ${\displaystyle O(r)}$ ${\displaystyle \mathbb {R} ^{r}}$

The Kolimes the sequence of inclusions

${\displaystyle Gr(1,2)\subset Gr(2,4)\subset \ldots \subset Gr(n,2n)\subset \ldots }$

is referred to as or . The terms are also used ${\displaystyle BGL(\mathbb {K} )}$ ${\displaystyle BO(\mathbb {K} )}$

${\displaystyle BO:=BO(\mathbb {R} ),BU:=BO(\mathbb {C} )}$.

The homotopy groups of this space can be calculated using Bott periodicity .

Schubert calculus

The cup product in the cohomology ring of the Graßmann manifolds can be determined using Schubert's calculus.

See also

• Boot manifold
• Flag manifold

Web links

• Grassmann manifold . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
• Eric W. Weisstein : Grassmann Manifold . In: MathWorld (English).