The rotation group $SO(3)$ may be mapped to a $2$-sphere by sending a rotation matrix to its first column. How can I describe the fibres of the map?

  • $\begingroup$ Any help here would be appreciated $\endgroup$ – user191141 Nov 24 '14 at 6:19
  • 1
    $\begingroup$ Jimmy, do not do this. You just edited two of your questions, the other being this question, to ask a complete different question identical at both posts. You were totally invalidating the answers given on both counts. $\endgroup$ – Mark Fantini Jan 5 '15 at 2:45

A $3 \times 3$ matrix $A$ is in $SO(3)$ iff its columns ${\bf a}, {\bf b}, {\bf c}$ form an oriented orthonormal basis of $\mathbb{R}^3$, so we can regard the fiber of the projection $\Pi: A \mapsto {\bf a}$ over a (unit-length) vector ${\bf a}_0$ as the set of such bases whose first entry is ${\bf a}_0$.

For any such basis, $({\bf b}, {\bf c})$ is an oriented orthonormal basis of the $2$-plane $\langle {{\bf a}_0} \rangle^{\perp}$, so we can identify the fiber $\Pi^{-1}({\bf a}_0)$ as the set of such bases. In fact, since $({\bf a}_0, {\bf b}, {\bf c})$ is oriented and orthonormal, $\bf b$ determines $\bf c$ via ${\bf c} = {\bf a}_0 \times {\bf b}$; thus, we can identify the fiber with the set of possible second basis elements $\bf b$, which is exactly the set of unit vectors in the plane $\langle {\bf a}_0 \rangle^{\perp}$, and which in turn is just a copy of the circle, $\mathbb{S}^1$.

Remark The universal cover of the space $SO(3)$ is $SU(2)$, which we can identify with $\mathbb{S}^3$. If we let $\pi$ denote the projection $SU(2) \to SO(3)$, then $\Pi \circ \pi$ is a bundle map $\mathbb{S}^3 \to \mathbb{S}^2$ with fiber $\mathbb{S}^1$, and this turns out to be precisely the classical Hopf Fibration.

  • $\begingroup$ Very well done indeed, sir, +1! $\endgroup$ – Robert Lewis Nov 24 '14 at 6:42
  • $\begingroup$ Cheers, Robert! $\endgroup$ – Travis Nov 24 '14 at 6:44
  • $\begingroup$ Indeed, Travis! $\endgroup$ – Robert Lewis Nov 24 '14 at 6:44

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy