R}^3$ and endowed with the Riemannian metric induced by the the Euclidean metric on ${\mathbb R}^3$. Let $\mathbb{T} = \{ z \in \mathbb{C}\colon |z| = 1 \}$ be the circle group with group operation given by complex multiplication and let $\rho(4)$ be the set of the $4$th roots of unity. A *cross* is an element of $C = \mathbb{T}/\rho(4)$. There is a canonical group isomorphism $R \colon C \rightarrow \mathbb{T}$ called the *representation map* given by $R([c]) = c^4,$ where $c$ is any representative member of the equivalence class $[c] \in C$. We will refer to $u = c^4$ as the *representation vector* for $[c]$. The *inverse representation map* $R^{-1} \colon \mathbb{T} \rightarrow C$ assigns $u \in \mathbb{T}$ to $R^{-1}(u) = \left[\sqrt[4]{u}\right]$, the equivalence class of the $4$th roots of $u$.

Let $T_p$ be the tangent space of $M$ at a point $p$. The disjoint union of all tangent spaces on $M$ is called the *tangent bundle* and is denoted $TM$. For each tangent space $T_p$ we select a coordinate basis, $\{\frac{\partial }{\partial x^1}|_p,\frac{\partial }{\partial x^2}|_p\}$. We also associate with each point $p$ of $M$ a space homeomorphic to $C$, which we call the *cross space* at $p$. We denote 