The fundamental equations of conformal submersions
The authors establish an analogy with the equations of O'Neill for Riemannian submersions. The method consists in attaching to each class of conformal submersions a unique Riemannian submersion, then applying appropriate projectors, taking the conformal invariant components of its fundamental equations. The method was previously used by the same authors for conformal immersions [``Conformal geometry of Riemannian submanifolds. I. Gauss equation; II. Codazzi and Ricci equations'' (Prepr. Univ. degli Studi di Roma (1992)]. After a brief introduction, in the second section the O'Neill equations for Riemannian submersions are rewritten in a slightly modified form. The algebraic preparation (decomposition of algebraic curvature tensors, conformal projectors), is made in section three. Section four introduces the two fundamental (1,2) tensors of a conformal submersion and shows that they practically determine it. In the final sections the five fundamental equations are derived. Conformal submersions are closely related to harmonic morphisms. Using different methods, \textit{S. Gudmundsson} found independently equations of this type [Ph. D. Thesis, University of Leeds (1992)] but in a different form.
scientific article; zbMATH DE number 510044