Distance on arrays of unit-quaternions

[mnb96]

Hello,
let's take the algebra of quaternions $\mathcal{C}\ell_{3,0}^+$ and consider only the sub-algebra of unit-quaternions.
When we wanted to define a https://www.physicsforums.com/showpost.php?p=2911801&postcount=6" we used the fact that unit-quaternions lie on the $\mathcal{S}^3$ sphere.

Now, what happens if we instead consider the whole set of finite "arrays" $(q_1,\ldots,q_n)$, in which each entry q_i is a unit-quaternion?

Is it possible that an n-tuple $(q_1,\ldots,q_n)$ represent a point on some manifold?
If so, how can we compute the geodesic distance between two n-tuples?

Thanks!


EDIT:
a simplified question could be: given a vector of unit complex numbers, can we consider this vector as a point of some manifold?

 

[fresh_42]

Hello,
let's take the algebra of quaternions $\mathcal{C}\ell_{3,0}^+$ and consider only the sub-algebra of unit-quaternions.
When we wanted to define a https://www.physicsforums.com/showpost.php?p=2911801&postcount=6" we used the fact that unit-quaternions lie on the $\mathcal{S}^3$ sphere.

Now, what happens if we instead consider the whole set of finite "arrays" $(q_1,\ldots,q_n)$, in which each entry q_i is a unit-quaternion?

Is it possible that an n-tuple $(q_1,\ldots,q_n)$ represent a point on some manifold?

Yes. We can simply consider direct products. Here are some equivalent presentations:
https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/

If so, how can we compute the geodesic distance between two n-tuples?

Depends on the metric considered.

Thanks!EDIT:
a simplified question could be: given a vector of unit complex numbers, can we consider this vector as a point of some manifold?

There is a difference between a vector and a point so your question is a bit unclear. Every point described via coordinates can be considered a point of a manifold. Which manifold might be the more interesting question as there is no unique answer.