Eigenvalues of harmonic almost submersions
  Maps between Riemannian manifolds which are submersions on a dense subset, are studied by means of the eigenvalues of the pull-back of the target metrics, the first fundamental form. Expressions for the derivatives of these eigenvalues yield characterizations of harmonicity, totally geodesic maps and biconformal changes of metric preserving harmonicity. A Schwarz lemma for pseudo harmonic morphisms is proved, using the dilatation of the eigenvalues and, in dimension five, a Bochner technique method, involving the Laplacian of the difference of the eigenvalues, gives conditions forcing pseudo harmonic morphisms to be harmonic morphisms. 
E. Loubeau and R. Slobodeanu