I am trying to understand, from a more formal point of view, the problem of gimbal lock. This led me to read about 3-torus, rotation group and real projective space.
I don't quite understand the difference between the 3-torus and the 3-sphere, for instance I read the 3-torus is defined as the product of S1 3 times, but then how about the SO3?
Is there a proof for the euler angles to be 3-torus and the quaternions to be 3-sphere and a proof for the 3-torus to not have a covering map on the real projective space while 3-sphere does?
I would appreciate both more intuitive explanations and more formal ones, I am not a mathematics student but an engineering student and I really like this topic so I am trying to get deeper into it.
Thanks!