Weak inverse problem of calculus of variations for geodesic mappings and relation to harmonic maps

TL;DR

This paper explores the relationship between geodesic and harmonic mappings, solving a weak inverse problem to reveal new insights into the connections and structures on fibred manifolds.

Contribution

It introduces a variational equation for geodesic mappings, showing the connection on the source need not be metric and that the target's metric can vary between fibers.

Findings

Connection on target manifold is metric.

Connection on source manifold need not be metric.

Target metric can change between fibers, related to source connection.

Abstract

In this paper, we study the relation between geodesic and harmonic mappings. Harmonic mappings are defined between Riemannian manifolds as critical points of the energy functional, on the other hand, geodesic mappings are defined in a more general setting (manifolds with affine connections). Using the well-established formalism of calculus of variations on fibred manifolds we solve the weak inverse problem for the equation of geodesic mappings and get a variational equation which is a consequence of the geodesic mappings equation. For the connection on the target manifold we get the expected result, that it is a metric connection. However, we find that the connection on the source manifold need not be metric. The interesting result is that the metric which induces the connection on the target manifold can change between fibres and these changes are related to the connection on the source manifold. These results hint onto a possibility for a more general structure on the fibred manifold, than usually assumed.