I have a number of related questions on how the quaternion exponential map can be defined, and in trying to read up on the topic I've gone down a rabbit hole in fields I have no background in. I apologize in advance if my terminology is off here; I'll try to summarize what I know to clarify my questions.
- Do both the set of imaginary quaternions and the set of unit quaternions parameterize $SU(2)$ via the exponential map? The classical axis-angle mapping shows that the unit quaternions do, so that seems sufficient proof in that case. For the imaginary (but not necessarily unit) case, I think that the exponential map should still work since the scaling of the quaternion relative to a unit pure quaternion provides the same information as a unit quaternion with nonzero real part.
- Is it true that each of the following three sets constitutes a Lie group: The set of unit quaternions, the set of imaginary quaternions, and the set of all nonzero quaternions?
- According to Wikipedia, the exponential map defined as above should coincide with the exponential map defined for the Riemannian manifold of the quaternions, but it's not clear to me how to define the appropriate metric for this to be the case. For the unit quaternions, the tangent space at 1 is just the imaginary quaternions; for non-unit quaternions, isn't the tangent space the same? But in that case, since the definition of the Lie exponential map says that the exponential of a quaternion is defined according to a subgroup whose tangent vector at the identity is q, if q is not pure imaginary then how do we define the exponential? I'm generally confused about how we can see the correspondence between the Lie and Riemannian exponential maps for quaternions. Is the tangent vector in this case just some scaled version of the imaginary part of the quaternion?
- Finally, if we can define the Riemannian exponential map as above to coincide with the Lie exponential map, the formula for quaternions that I have seen suggests it should be defined as $q*exp(d)$. What does this mapping mean, and how does it relate to the Riemannian exponential map? Also, is this definition still meaningful for non-unit quaternions?