Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 115 / 3 / d

Table of contents
- Solution d

Solution

Let

J = (01 - 1 0) .

The vertical space is spanned by the fundamental vector

(y, - x)

corresponding to

J \in o (2)

A_{(x, y)} (u, v) = ⟨ u, y ⟩ J = - ⟨ v, x ⟩ J .

It sends

(y, - x)

to

J

, is

O (2)

-equivariant, and therefore is a principal connection. Its kernel consists exactly of those

(u, v)

for which

⟨ u, x ⟩ = ⟨ u, y ⟩ = ⟨ v, x ⟩ = ⟨ v, y ⟩ = 0.

These are precisely the velocities satisfying the stated horizontality condition. Every tangent vector has the unique decomposition

(u, v) = ⟨ u, y ⟩ (y, - x) + ((u, v) - ⟨ u, y ⟩ (y, - x)),

into vertical and horizontal parts, proving uniqueness. This is the Canonical principal connection on the Stiefel bundle over a Grassmannian.

 Ancestors (10)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version