Stiefel manifold

In mathematics parameterize boots -manifolds, named after Eduard Stiefel, the bases of subspaces of a vector space.

Definition

Be, or (skew - ) field of real, complex or quaternionic numbers and let one -dimensional vector space. Be.

Then the boots -manifold is defined as the set of all tuples of linearly independent vectors.

Action of the linear group

The group acts transitively on with stabilizer, so we obtain a bijection

In fact, even the orthogonal or unitary groups act already transitive and gives bijections

Topology

One uses the bijection to define a topology with which the bijection a homeomorphism is. With this topology, the manifolds to the following dimensions:

Equivalent can also define the topology by the canonical identification of a subspace of.

Principal bundle over the Grassmann manifold

The Grassmann manifold is the set of -dimensional subspaces of.

Each tuple of linearly independent vectors can be assigned to the subspace generated by it, in this way one defines a projection

The projections are defined as the principal bundle

Boots -manifolds in discrete mathematics

The graph homomorphisms complex is homeomorphic to the Stiefel manifold ( Csorba 's conjecture, proved by Schultz ).

Documents

• Differential topology