The Green functions in curved spacetime

The theory of a free scalar field with conformal coupling in curved spacetime with some special metrics is considered. The integral representations for the Green function G approximately _in(0 mod T phi (x) phi (x') mod 0)_in in the form of integrals with Schwinger-De Witt kernel over contours in the complex plane of proper time are obtained. It is shown how the transitions from a unique Green function in Euclidean space to different Green functions in Minkowski space and vice versa can be carried out.