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Left group actions involving SO(3) and the 2-sphere

  1. Oct 25, 2011 #1
    the group of proper orthogonal transformations SO(3) acts transitively on the 2-sphere S2.

    show that the isotropy group of any vector r is isomorphic to SO(2) and find a bijective correspondence between the factor space SO(3)/SO(2) and the 2-sphere such that SO(3) has identical left actions on these two spaces.

    for the first part i have to show that the set of all 3x3 orthogonal matrices A such that Ar = r is isomorphic to the set of all 2x2 orthogonal matrices. so each matrix in SO(3) represents a rotation of some kind. but it seems to me that the only 3x3 orthogonal matrices that satisfy Ar = r are those that represent 360 degree rotations around some axis and it seems to me that set of all 2x2 orthogonal matrices contains more than just full rotations around some axis. so i am having trouble coming up with a bijection between the two sets.

    for the second part i am confused about what they mean by identical left actions. i think they mean the same function on both which makes sense since we will be showing that SO(3)/SO(2) and the 2 sphere are in bijection. but doesn't it have to be an isomorphism as well to consider them to be the same group? i am also having trouble coming up with a bijection for the second part. could someone offer some hints to help me continue with this problem? thanks!
     
  2. jcsd
  3. Oct 26, 2011 #2
    for the second part

    You have a transitive action of [itex]G=SO(3)[/itex] on the set [itex]A=S^2[/itex] and [itex]G_r[/itex] the isotropy group of [itex]r[/itex].

    You know that [itex]G[/itex] acts on [itex]G/G_r[/itex] - by [itex]g\cdot xG_r = gxG_r[/itex].

    You have to define a bijection [itex]f: G/G_r \to A[/itex] such that "[itex]G[/itex] has identical action on [itex]A[/itex] and on [itex]G/G_r[/itex]".

    I think that this means that you have to define a bijection [itex]f[/itex] such that [itex]f(w\cdot xG_r) = w \cdot f(xG_r)^{(1)}, \,\, \forall w\in G, \,\, \forall xG_r \in G/G_r.[/itex]

    You can define [itex]f(xG_r)=x\cdot r[/itex] and then to prove that [itex]f[/itex] is well-defined, [itex]1-1[/itex], onto and finally that (1) holds.

    For the "onto" part you must use that [itex]G[/itex] acts transitively on [itex]A[/itex].

    Notice that [itex]G/G_r[/itex] is not a group so [itex]f[/itex] is not a group isomorphism - [itex]f[/itex] is just a bijection between two sets.
     
  4. Oct 27, 2011 #3
    for the first part i have to show that SO(2) is isomorphic with the set of matrices A in SO(3) such that Ar = r for some fixed unit vector r.

    since i know the determinant of any A in SO(3) must be 1, i know one of its eigenvalues must be 1 and the other two must multiply to 1. the unit vector r must be an eigenvector of A with eigenvalue 1.

    i am claiming that for every 2x2 matrix with determinant 1 in SO(2) i can associate it with a unique 3x3 matrix in SO(3) such that this 3x3 matrix has the same eigenvalues as the 2x2 matrix along with 1. In addition, this matrix must have the property that Ar = r for some fixed unit vector r.

    i wasn't sure how to describe this mapping using a formula but i need to prove that it is bijective and a homomorphism. i defined the function f: SO(2) -> SO(3) such that f takes any 2x2 matrix in SO(2) to a unique 3x3 matrix in SO(3). but without a concrete formulation for it i don't know how to show that its an isomorphism. is this the right way to go?
     
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