The curvature of the domain on various combinations of horizontal and vertical vectors is calculated for a horizontally conformal submersion and for a (submersive) harmonic morphism. It is related to the dilation and the curvature of the codomain, and local and global non-existence results are deduced for maps with or without critical points. It is shown that any harmonic morphism with totally geodesic fibres — defined on a Euclidean space with values in a manifold of dimension not equal to two — is orthogonal projection to a subspace followed by a surjective homothetic covering. Together with Theorem 6.7.3, this gives a complete picture of the globally defined harmonic morphisms with totally geodesic fibres on Euclidean space.
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