Rotate a three-sphere so every point moves in R^4?

[Spinnor]

If we rotate a two-sphere in R^3 so it stays centered about the origin then two points don't move.

If we rotate a three-sphere in R^4 while it stays centered about the origin can it be done so that all points move?

Does this have anything to do with the fact the S^3 is parallelizable while S^2 is not?

Thanks for any help!

 

[WWGD]

Rotations are linear maps so that you can represent them as matrices after you choose a basis. Then solve Mx=x, where M is your matrix.

 

[lavinia]

If we rotate a two-sphere in R^3 so it stays centered about the origin then two points don't move.

If we rotate a three-sphere in R^4 while it stays centered about the origin can it be done so that all points move?

Does this have anything to do with the fact the S^3 is parallelizable while S^2 is not?

Thanks for any help!

If you view R^4 as C^2, complex 2 space, then the 3 sphere is all pairs of complex numbers $$(se^{iθ}, re^{iα} )$$ such that $$r^2 + s^2 =1$$
You can see the answer to both of your questions from this representation.

 

[WWGD]

Every odd dimensional sphere can be rotated without having a fixed point. Just use the standard matrix representation of the rotation and do it so that none of the eigenvalues is equal to 1. So it does not have to see with parallelizability.

 

[Spinnor]

Sorry for the delay in thanking you both.