Quaternion analyticity and conformally K�hlerian structure in Euclidean gravity

Starting from the fact that the d=4 Euclidean flat spacetime is conformally related to the Kahler manifold H2×S2, we show the Euclidean Schwarzschild metric to be conformally related to another Kahler manifold M2×S2 with M2 being conformal to H2 in two dimensions. Both metrics which are conformally Kahlerian, are form-invariant under the infinite parameter Fueter group, the Euclidean counterpart of Milne's group of clock regraduation. The associated Einstein's equations translate into Fueter's quaternionic analyticity. The latter leads to an infinite number of local continuity equations.