Eigenvalues of harmonic almost submersions

TL;DR

This paper investigates the eigenvalues of the pull-back metric in harmonic almost submersions, providing new characterizations of harmonicity, and establishing conditions under which pseudo harmonic morphisms are harmonic.

Contribution

It introduces eigenvalue-based characterizations for harmonicity and totally geodesic maps, and proves a Schwarz lemma for pseudo harmonic morphisms using eigenvalue dilatation.

Findings

Eigenvalues characterize harmonicity and totally geodesic maps.

A Schwarz lemma for pseudo harmonic morphisms is established.

Conditions are identified where pseudo harmonic morphisms become harmonic morphisms.

Abstract

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.