Harmonic G-structures

TL;DR

This paper investigates the energy functional on G-structures of Riemannian manifolds, deriving critical point conditions and exploring harmonic G-structures, especially focusing on the case G=U(n) for even-dimensional manifolds.

Contribution

It introduces the first and second variation formulas for the energy functional on G-structures and characterizes harmonic G-structures using intrinsic torsion, with a focus on the U(n) case.

Findings

Derived variation formulas for the energy functional.

Characterized critical points via intrinsic torsion.

Analyzed harmonic almost Hermitian structures and maps.

Abstract

For closed and connected subgroups G of SO(n), we study the energy functional on the space of G-structures of a (compact) Riemannian manifold M, where G-structures are considered as sections of the quotient bundle O(M)/G. Then, we deduce the corresponding first and second variation formulae and the characterising conditions for critical points by means of tools closely related with the study of G-structures. In this direction, we show the role in the energy functional played by the intrinsic torsion of the G-structure. Moreover, we analyse the particular case G=U(n) for even-dimensional manifolds. This leads to the study of harmonic almost Hermitian manifolds and harmonic maps from M into O(M)/U(n).