Vertical Ricci curvature of totally geodesic submersions

LaTeX source and figure script of the paper Vertical Ricci curvature of totally geodesic submersions. A Riemannian submersion with compact, connected, totally geodesic fibres and nonpositive Ricci curvature on vertical vectors has vanishing O'Neill tensor A and is locally a Riemannian product; a complete total space has universal cover the product of the universal covers of a fibre and the base. Compact Ricci-flat manifolds, in particular Calabi-Yau, hyperkahler, G2 and Spin(7) manifolds with irreducible holonomy, admit no such submersion onto a lower-dimensional base. The proof shows that the vertical bracket of basic fields is a Killing field of each fibre and applies Bochner's formula; the appendices derive every formula used and compute the Hopf and Heisenberg examples.