# A characterization of harmonic sections and a Liouville theorem

## Simão Stelmastchuk

Address: Departamento de Matemática, Universidade Estadual do Paraná, 84600-000 – União da Vitória - PR, Brazil

E-mail: simnaos@gmail.com

Abstract: Let $P(M,G)$ be a principal fiber bundle and $E(M,N,G,P)$ an associated fiber bundle. Our interest is to study the harmonic sections of the projection $\pi _{E}$ of $E$ into $M$. Our first purpose is give a characterization of harmonic sections of $M$ into $E$ regarding its equivariant lift. The second purpose is to show a version of a Liouville theorem for harmonic sections of $\pi _{E}$.

AMSclassification: primary 53C43; secondary 55R10, 58E20, 58J65, 60H30.

Keywords: harmonic sections, Liouville theorem, stochastic analysis on manifolds.

DOI: 10.5817/AM2012-2-149