Existence conditions for conformal submersions with totally umbilical fibers

Conformal submersions are a natural generalisation of Riemannian submersions, introduced by O'Neill (1966). Some existence conditions for conformal submersions were obtained by Gudmundsson (1990) through comparison of curvatures of the domain and the image of the submersion. Baditoiu and Ianus (2002) derived such conditions for a Riemannian submersion with totally umbilical fibers from the formula for mixed scalar curvature of its domain.We show how in case of a conformal submersion with totally umbilical fibers these two approaches yield partial differential equations for the dilation of such submersion. From these equations we obtain some obstructions to existence of conformal submersions with totally umbilical and totally geodesic fibers.