First of all I have to say that I am not very competent in topology, so please try not to use too obscure terms in your answers and pardon me for my explanations that may not be very rigorous.
I am using a 3-manifold that is almost (you'll see why I say almost in a few lines) a 3-sphere defined on $(\psi, \theta, \phi)$ that are respectively defined on $[0 ; \pi/2], [0 ; \pi/2], [0 ; 2\pi]$. So it is a some sort of a quarter 3-sphere.
But this manifold has an important property: for every possible set of coordinates $(\psi, \theta, \phi)$, the value at this position will be equal to the one at the position $(\theta, \psi, \phi)$. In other words there is a symmetry between the coordinates $\psi$ and $\theta$.
I am struggling to figure out what is the consequence of this on the shape of the 3-manifold, but this is more by curiosity. The real question is: since there is a symmetry, we have redundancy of information. Is there a possibility to parametrize the manifold to a new set of coordinates (or to find an equivalent manifold) to simplify it and avoid redundancy?
Any help, or any advice, would be appreciated.
EDIT : Here is a first attempt.
If this symmetry property is true, we can propose the following parameterization $(u,v,w)$, with $u \in [0, \pi/2]$, $v \in [0, u]$ and $w \in [0, 2\pi]$, with :
$$\begin{cases}u = \psi \text{ if } \theta < \psi \text{ else } \theta\\ v = \theta \ \text{ if } \theta < \psi \text{ else } \psi\\ w = \phi\end{cases}$$
This will avoid redundancy. But what will be the consequences of this?
