A NOTE ON CONSTANT CURVATURE SOLUTIONS IN CYLINDRICALLY SYMMETRIC METRIC f(R) GRAVITY

In the previous work we introduced a new static cylindrically symmetric vacuum solution in Weyl coordinates in the context of the metric f(R) theories of gravity.(1) Now we obtain a two-parameter family of exact solutions which contains a cosmological constant and a new parameter as beta. This solution corresponds to a constant Ricci scalar. We proved that in f(R) gravity the constant curvature solution in cylindrically symmetric cases is only one member of the most generalized Tian family in GR. We show that our constant curvature exact solution is applicable to the exterior of a string. The sensibility of stability under initial conditions is discussed.