Is there a compact cylindrical universe solution to the Einstein equation, with space homogeneity, without using "artificial" periodic boundaries? I'm expecting a metric of the following shape: \begin{equation}\tag{1} ds^2 = f^2(t, r) \, dt^2 - g^2(t, r)(dx^2 + dy^2) - b^2(t) \, dz^2, \end{equation} where $f(t, r)$, $g(t, r)$ and $b(t)$ are some functions and $r = \sqrt{x^2 + y^2}$. I'm also expecting $f(t, r) = 1$ and $g(t, r) = a(t) \, g(r)$. The cylindrical planar space section (coordinates $x$ and $y$) should be homogeneous and compact.
I'm thinking of something similar to a 2D version of the spherical closed universe in cosmology ($k = +1$), with an extra space dimension, of FLRW metric \begin{equation}\tag{2} ds^2 = dt^2 - \frac{a^2(t)}{(1 + r^2 / 4)^2}(dx^2 + dy^2 + dz^2) - b^2(t) \, du^2. \end{equation}
The metric (1) I'm looking for would have an homogeneous 3D space, would be 2D isotropic in the cylindrical plane (in coordinates $x$ and $y$), but would not be isotropic in 3D.
EDIT 2 (I've fixed some important mistakes): If I assume $f(t, r) = 1$ and $g(t, r) = a(t) \, g(r)$, I could compute the Christoffel symbols and get these (all the other symbols are trivially 0. Here: $i, j, k = 1, 2$ for coordinates $x$ and $y$): \begin{align} \Gamma_{ij}^0 &= a \dot{a} g^2 \, \delta_{ij}, \\[12pt] \Gamma_{33}^0 &= b \dot{b}, \\[12pt] \Gamma_{0 j}^i &= \frac{\dot{a}}{a} \, \delta_{ij}, \\[12pt] \Gamma_{jk}^i &= \frac{1}{g}(\partial_j g \, \delta_{ik} + \partial_k g \, \delta_{ij} - \partial_i \, g \, \delta_{jk}), \\[12pt] \Gamma_{03}^3 &= \frac{\dot{b}}{b}. \end{align} Then, I compute the Riemann components (all other components are 0): \begin{align} R^0_{i 0 j} &= a \ddot{a} \, g^2 \, \delta_{ij}, \\[12pt] R^0_{303} &= b \ddot{b}, \\[12pt] R^i_{33j} &= -\, \frac{\dot{a} \dot{b} b}{a} \, \delta_{ij}, \\[12pt] R^i_{jkl} &= (\dot{a}^2 g^2 + \mathcal{Q}(r))(\delta_{ki} \, \delta_{lj} - \delta_{li} \, \delta_{kj}) + \text{anisotropic terms}. \end{align} The anisotropic terms cancel if \begin{equation}\tag{3} \frac{g^{\prime \prime}}{g} = \frac{g^{\prime}}{g \, r} + 2 \frac{g^{\prime 2}}{g^2}, \end{equation} which implies the solution \begin{equation}\tag{4} g(r) = \frac{A}{1 + B \, r^2}, \end{equation} where $A$ and $B$ are constants (they can be absorbed into the coordinates, so $A = 1$ and $B = -1, 0, 1$ like the parameter $k$ in cosmology). Then \begin{equation}\tag{5} \mathcal{Q}(r) = 4 B \, g^2. \end{equation} I then compute the Einstein tensor (again: $i, j = 1, 2$ only): \begin{align} G_{00} &= -\, \Big( 2 \frac{\dot{a} \dot{b}}{a b} + \frac{\dot{a}^2}{a^2} + \frac{4 B}{a^2} \Big), \tag{6} \\[12pt] G_{ij} &= -\, \Big( \frac{\ddot{a}}{a} + \frac{\ddot{b}}{b} + \frac{\dot{a} \dot{b}}{a b} \Big) \, g_{ij}, \tag{7} \\[12pt] G_{33} &= -\, \Big( 2 \frac{\ddot{a}}{a} + \frac{\dot{a}^2}{a^2} + \frac{4 B}{a^2} \Big) \, g_{33}. \tag{8} \end{align} These components are similar (but have some differences) with what we find in the usual spherical FLRW cosmology.
I guess that I need to introduce some anisotropic pressure ($\sigma = p_{\perp}$ in the $x$-$y$ plane and $p = p_{\parallel}$ along the $z$ axis). Is there any interesting paper on anisotropic-cylindrical cosmologies?
