An exact solution to the heat equation in curved space is a much sought after; this report presents a derivation wherein the cylindrical symmetry of the metric gμν in 3 + 1 dimensional curved space has a pivotal role. To elaborate, the spherically symmetric Schwarzschild solution is a staple of textbooks on general relativity; not so perhaps, the static but cylindrically symmetric ones, though they were obtained almost contemporaneously by H. Weyl, Ann. Phys. Lpz. 54, 117 (1917) and T. Levi-Civita, Atti Acc. Lincei Rend. 28, 101 (1919). A renewed interest in this subject in C.S. Trendafilova and S.A. Fulling, Eur.J.Phys. 32, 1663(2011) - to which the reader is referred to for more references - motivates this work, the first part of which (cf.Kamath, PoS (ICHEP2016) 791) reworked the Antonsen-Bormann idea - arXiv:hep-th/9608141v1 - that was originally intended to compute theheat kernel in curved space to determine - following D.McKeon and T.Sherry, Phys. Rev. D 35, 3584 (1987) - the zeta-function associated with the Lagrangian density for a massive real scalar field theory in 3 + 1 dimensional stationary curved space to one-loop order, the metric for which is cylindrically symmetric. Using the same Lagrangian density the second part reported here essentially revisits the second paper by Bormann and Antonsen - arXiv:hep 9608142v1 but relies on the formulation by the author in S. G. Kamath, AIP Conf.Proc.1246, 174 (2010) to obtain the Green's function directly by solving a sequence of first order partial differential equations that is preceded by a second order partial differential equation.
Cylindrical symmetry: II. The Green's function in 3+ 1 dimensional curved space