The Ortogonal Representation of SU2

[SNOOTCHIEBOOCHEE]

 

[SNOOTCHIEBOOCHEE]

Homework Statement

The rotation group SO3 may be mapped to the 2 sphere by sending a rotation matrix to its first column. Describe the fibres of this map

Homework Equations

SO3 are the special matrices P such that PP^t=I and they are 3x3 matrices with det P =1

The Attempt at a Solution

ok so we just want to describe the inverse image of an element. I know the image of the element is a matrix which corresponds to some rotation is sent to a 2-sphere. But i don't really know how to describe this formally or in the other direction.i know we have 3x3 matrices with det =1 st. pp^t

so our map (phi):SO₃ ---> S²

but don't know where to go from here.

 

[Dick]

Think of them as what they are, rotations. If two rotations R1 and R2 have the same first column then they map (1,0,0)=e1 to the same vector, call it v (the first column of the matrix). They both map e2 and e3 to vectors orthogonal to v. They must differ only by a rotation in the plane orthogonal to v, right?