Boot manifold

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

definition

Be or the (skew) field of real, complex or quaternionic numbers and be a -dimensional vector space. Be . ${\displaystyle \mathbb {K} =\mathbb {R} ,\mathbb {C} }$ ${\displaystyle \mathbb {H} }$ ${\displaystyle V=\mathbb {K} ^{n}}$ ${\displaystyle n}$ ${\displaystyle \mathbb {K} }$ ${\displaystyle 0\leq k\leq n}$

Then the Stiefel manifold is defined as the set of all -tuples of orthonormal vectors. ${\displaystyle V_{k}(\mathbb {K} ^{n})}$ ${\displaystyle k}$

The non-compact Stiefel manifold is defined as the set of -tuples of linearly independent vectors. The inclusion of in the non-compact Stiefel manifold is a homotopy equivalence . ${\displaystyle k}$ ${\displaystyle V_{k}(\mathbb {K} ^{n})}$

Effect of the linear group

The group acts transitively on the non-compact Stiefel manifold with stabilizer , so you get a bijection with ${\displaystyle GL(n,\mathbb {K} )}$ ${\displaystyle GL(k,\mathbb {K} )}$

${\displaystyle GL(n,\mathbb {K} )/GL(k,\mathbb {K} )}$.

On the Stiefel manifold , even the orthogonal or unitary groups already act transitive and one obtains bijections ${\displaystyle V_{k}(\mathbb {K} ^{n})}$

${\displaystyle {\begin{aligned}V_{k}(\mathbb {R} ^{n})&\cong {\mbox{O}}(n)/{\mbox{O}}(n-k)\\V_{k}(\mathbb {C} ^{n})&\cong {\mbox{U}}(n)/{\mbox{U}}(n-k)\\V_{k}(\mathbb {H} ^{n})&\cong {\mbox{Sp}}(n)/{\mbox{Sp}}(n-k).\end{aligned}}}$

topology

The above bijections are used to define a topology with which the bijection becomes a homeomorphism . With this topology, they become manifolds of the following dimensions: ${\displaystyle V_{k}(\mathbb {K} ^{n})}$ ${\displaystyle V_{k}(\mathbb {K} ^{n})}$

${\displaystyle \dim V_{k}(\mathbb {R} ^{n})=nk-{\frac {1}{2}}k(k+1)}$

${\displaystyle \dim V_{k}(\mathbb {C} ^{n})=2nk-k^{2}}$

${\displaystyle \dim V_{k}(\mathbb {H} ^{n})=4nk-k(2k-1).}$

The topology can also be defined equivalently by the canonical identification of with a subspace of . ${\displaystyle V_{k}(\mathbb {K} ^{n})}$ ${\displaystyle \mathbb {K} ^{nk}}$

Principal bundle over the Graßmann manifold

The Graßmann manifold is the set of -dimensional subspaces of the . ${\displaystyle G_{k}(\mathbb {K} ^{n})}$ ${\displaystyle k}$ ${\displaystyle \mathbb {K} ^{n}}$

Each tuple of linearly independent vectors can be assigned the sub-vector space generated by it, in this way a projection is defined ${\displaystyle k}$

${\displaystyle V_{k}(\mathbb {K} ^{n})\rightarrow G_{k}(\mathbb {K} ^{n})}$.

The projections so defined are principal bundles

${\displaystyle {\begin{aligned}\mathrm {O} (k)&\to V_{k}(\mathbb {R} ^{n})\to G_{k}(\mathbb {R} ^{n})\\\mathrm {U} (k)&\to V_{k}(\mathbb {C} ^{n})\to G_{k}(\mathbb {C} ^{n})\\\mathrm {Sp} (k)&\to V_{k}(\mathbb {H} ^{n})\to G_{k}(\mathbb {H} ^{n}).\end{aligned}}}$

Boot Manifolds in Discrete Mathematics

The graph-homomorphism complex is homeomorphic to the Stiefel manifold (Csorba's conjecture, proven by Schultz). ${\displaystyle Hom(C_{5},K_{n})}$ ${\displaystyle V_{2}(\mathbb {R} ^{n-1})}$

supporting documents

1. ↑ Small models of graph coloring manifolds and the Stiefel manifold Hom (C ₅ , K _n ) pdf