Seamed Riemann Surfaces

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 Consider a seamed orbifold Riemann surface constructed as follows. Start with the face poset of a three-dimensional cube. Let $\mathcal A_3$ denote the set of antipodal vertex pairs $$a=\{v,v^*\}.$$ To each such pair $a\in\mathcal A_3$, associate a spherical $2$-orbifold $$S^2(\alpha,\alpha)$$ whose two cone points are attached to the antipodal vertices $v$ and $v^*$. Equivalently, and more naturally from the complex-analytic point of view, we regard these spherical orbifolds as copies of $\mathbb{CP}^1$ and glue them along a common seam complex $\Gamma$. Thus the complex foam is $$\mathcal I_{\mathbb C}=\left(\bigsqcup_{a\in\mathcal A_3}^{4}\mathbb{CP}^1_a\right)\big/\!\sim_\Gamma .$$ The cubical scaffold should be understood primarily as an organizational device. Rather than constructing the foam by gluing disks one at a time, as in more traditional constructions of seamed Riemann surfaces or Klein foams, the cube records the incidence structure in advance: whenever two sheets o...
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A description of $\zeta-$space

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I would like to attempt to define a mathematical space I call $\zeta-$space.  $\zeta-$space has a description as the transversal intersection of two isometric pseudo-Riemannian manifolds (Lorentzian submanifolds) both equipped with the Lorentz metric. It is somewhat laborious to describe and derive $\zeta$ because it is not a standard approach and $\zeta$ does not have a clean representation so to speak. However it is a fundamentally important mathematical and physical object in its own right and should be studied.

 I define $\zeta$ in the following way:

 $\zeta:=\{\varphi_S \} \cap \{ \varphi_T \}$

 Under this first level description, $\zeta$ is the intersection between two class structures denoted as $\varphi_S$ and $\varphi_T.$ These class structures can be thought of as a family of functions and also as lines of constant time and constant space respectively:

 $$ \varphi_S(x) = e^{\frac{S}{\log(x)}} $$

 $$ \varphi_T(x) = e^{\frac{T}{\log(1-x)}}$$

 Here I view $S,T$ as fundamental generators. That is, they completely generate the space. Here is how I derived them and began to think of them as generators:

 I set $\varphi_S(x)=x$ and solved for $S.$ I got $S=\{\log^2(x) : x \in \Bbb R \cap (0,1) \}.$ However I thought it better to replace $x$ with some parameter $s.$ So, $S=\{\log^2(s) : s \in \Bbb R \cap (0,1) \}.$ Likewise I set $\varphi_T(x)=1-x$ and solved for $T.$

 Here we have a visualization of $\zeta$ showing the overlapping class structures $\{ \varphi_S\}$ and $\{\varphi_T\}:$

$\zeta-$space as the intersection of two Lorentzian submanifolds


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