Consider a cylindrical mass in empty space. In general relativity, do the space-time dimensions curve about it in a completely symmetrical manner? Consider a cylindrical mass in empty space oriented along the x1 direction.  In general relativity, do the space-time dimensions (x1, x2, x3, x4) curve about the mass in a completely symmetrical manner?
 A: You're asking about the Ricci curvature tensor $R_{\mu\nu}$ provided the stress-energy tensor $T_{\mu\nu}$ (a local mass), i.e., a solution to Einstein's equations.  The easy way to solve this is to realize since flat space is isotropic and homogeneous, only a spherical mass will produce a symmetric curvature (assuming the object is inertial).  So the answer is no.
A: You need to be more precise about what you mean by do the space-time dimensions curve about it in a completely symmetrical manner?.
The cylinder has axial symmetry. Suppose we use cylindrical polar coordinates $r$, $\phi$ and $z$, then the axial symmetry means the stress-energy tensor is independent of the angle $\phi$. That means the metric will also be independent of $\phi$, which essentially means the spatial curvature is independent of the angle. If this is what you meant by completely symmetrical then the answer to your question is yes.
If the cylinder has an infinite length then the curvature will likewise be independent of $z$, so it will depend only on the distance from the axis.
