**Polar Coordinates:**

$$z\_0 = -\mathbf{i}\sin\theta\cosh\varphi,\tag{1a}$$

$$z\_1 = -\mathbf{i}\sin\theta\,\sinh\varphi,\qquad(\theta,\,\boldsymbol{\uprho}\in\mathbb{C}),\tag{1b}$$

$$z\_2 = \cos \theta.\tag{1c}$$

If *θ* = *u* (*u* ∈ R) and *ϕ* = i*ψ* (*ψ* ∈ R), (1) reads

$$\mathbf{x}\_0 = -i\sin\mu\cos\psi\_\prime \tag{2a}$$

$$\mathbf{x}\_1 = \sin \mu \sin \psi,\qquad \qquad (\mathbf{u}, \boldsymbol{\psi} \in \mathbb{R}),\tag{2b}$$

$$x\_2 = \cos u.\tag{2c}$$

It can be easily verified that: −*x*<sup>2</sup> 0 + *x*2 1 + *x*2 2 = sin<sup>2</sup> *u* + cos<sup>2</sup> *u* = 1 that is, one obtains as a real submanifold of the complex one-sheeted hyperboloid *X*(*c*) 2 the *Euclidean sphere* S2 = (iR × R<sup>2</sup>) ∩ *X*(*c*) 2 (iR referring to the coordinate *z*0 of *z*).

Similarly, if *θ* = i*v* (*v* ∈ R) and *ϕ* = *ψ* (*ψ* ∈ R), (1) becomes

$$\mathbf{x}\_0 = \sinh \boldsymbol{\upsilon} \cosh \boldsymbol{\psi},\tag{3a}$$

$$\mathbf{x}\_1 = \sinh \upsilon \sinh \psi,\qquad(\upsilon, \psi \in \mathbb{R}),\tag{3b}$$

*x*2 = cosh *v*. (3c)

In this case, it is easily verified that: −*<sup>x</sup>*20 + *x*21 + *x*22 = cosh<sup>2</sup> *v* − sinh<sup>2</sup> *v* = 1. Equation (3) describes only the subset *<sup>X</sup>*+2 .= {*x* ∈ *X*2 : *x*2 1} of the real one-sheeted hyperboloid *X*2, which is precisely what we need for the current analysis. Finally, let *X* -+ denote the (real) meridian lying in the plane *x*1 = 0 with equation: *x*20 − *x*22 = −1.
