In S. Hawking's and G. Ellis's book "The Large-Scale Structure of Space Time", they discuss the notion of inextendible Lorentz manifolds $(M,g)$ (see chapter 3.1). A Lorentz manifold is called inextendible if there is no isometric embedding into a larger Lorentz manifold.
As an example of an extendible manifold, they name the two-dimensional real plane $(M_1,g_1)$ with the $x$-axis removed between $x_1=-1$ and $x_1=+1$. The obvious extension is the "complete" real plane, where the removed points are simply replaced,
but one could also extend it by taking another copy $(M_2 , g_2)$ of the space, and identifying the bottom side of the $x_1$-axis for $|x_1| < 1$ with the top side of the $x_2$-axis for $|x_2| < 1$, and also identifying the top side of the $x_1$-axis for $|x_1| < 1$ with the bottom side of the $x_2$ -axis for $|x_2| < 1$. The resultant space $(M_3, g_3)$ is inextendible, but not complete as we have left out the points $x_1 = ± 1, y_1 = 0$. We cannot put these points back in because we were perverse enough to extend the top and bottom sides of the $x$-axis on different sheets.
I do not fully understand this construction.
- Is $(M_2,g_2)$ a copy of $(M_1, g_1)$ or a copy of the "complete" Euclidean plane?
- Are $x_1$ and $x_2$ the $x$-coordinates of $M_1$ and $M_2$, respectively?
- What do they mean by "the bottom side" and "the top side"? Does this mean that $-1<x_1\leq 0$ is identified with $0\leq x_2<1$ via $t\mapsto -t$?
- What happens to the points of $(M_1,g_1)$ and $(M_2,g_2)$ away from the $x$-axes. Are they identified as well?