Grassmannian

Grassmann manifolds (also Grassmann manifolds ) are a fundamental concept in mathematics both differential geometry and algebraic geometry. You parameterize the subspaces of a vector space. They are named after Hermann Grassmann.

Definition

Be a vector space over a field. Then called

The set of r-dimensional subspaces of. When n -dimensional is to be also referred to as

Effect of the orthogonal / unitary and linear group

In the case of the orthogonal group acts

On by

The action is transitive, the stabilizers are conjugate to

One thus gets a bijection between and the homogeneous space

In the case of the unitary group acts transitively and provides a bijection of the Grassmann manifold with

Are obtained analogously for arbitrary body is a bijection between and

Topology

As a real Grassmann manifold ( the r- dimensional subspaces in ) is called with the through identification with

Given topology.

Being a complex Grassmann manifold is called accordingly

The canonical inclusion induces an inclusion. One defines

As the projective limit with the limit topology.

Tautological bundle

Be the projective limit with respect to the canonical inclusions and define

Then the projection on the first factor, a vector bundle

Which is referred to as tautological or universal r -dimensional vector bundles.

Classifying map

There is every r -dimensional vector bundle is a continuous map

So that the pullback of the tautological bundle is below.

In the case of the tangent bundle of a differentiable manifold one has the following explicit description of the classified image: After embedding theorem of Whitney, one can assume that a submanifold is one. The tangent plane at a point is then of the form

For a subspace. The assignment

Defines a continuous map

And it can be shown that

Is.

Classifying space for principal bundles

The Grassmann manifold is the classifying space for principal bundles with structure groups. And thus for principal bundles with structure group, for it is because the inclusion is a homotopy equivalence, every bundle can reduce the structure group. Thus:

The canonical projection from the Stiefel manifold according to which each Repere maps onto the subspace generated by them, is the universal bundle. ( The tautological bundle arises from the universal bundle as an associated vector bundle with the canonical action of on the vector space. )

The colimit of the sequence of inclusions

Is designated as, or. Common are also the designations

Using Bott periodicity can compute the homotopy groups of this space.

Schubert calculus

The cup product in the cohomology ring of the Grassmann manifolds can be determined by the Schubert calculus.