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We approximate a Euclidean version of a D+1 dimensional manifold with abifurcate Killing horizon by a product of a two-dimensional Rindler space and aD-1 dimensional manifold M. We obtain approximate formulas for the Greenfunctions. We study the behaviour of Green functions near the horizon and theirdimensional reduction. We show that if M is compact then the massless minimallycoupled quantum field contains a zero mode which is a conformal invariant freefield on R^2. Then, the Green function near the horizon can be approximated bythe Green function of the two-dimensional quantum field theory. The correctionterm is exponentially small away from the horizon. If the volume of a geodesicball is growing to infinity with its radius then the Green function cannot beapproximated by a two-dimensional one.

Green functions and Euclidean fields near the bifurcate Killing horizon