## Relatório de pesquisa 29/09

Stochastic Characterization of Harmonic Sections and a Liouville Theorem, Simão Stelmastchuk, submitted October 22, 2009.

Abstract

Let $P(M,G)$ be a principal fiber bundle and $E(M,N,G, P)$ be an
associate fiber bundle. Our interest is to study harmonic sections of
the projection $\pi_E$ of $E$ into $M$. Our first purpose is to give a
stochastic characterization of harmonic section from $M$ into $E$ and a
geometric characterization of harmonic sections with respect to its
equivariant lift. The second purpose is to show a version of Liouville
theorem for harmonic sections and to prove that section $M$ into $E$ is
a harmonic section if and only if it is parallel.

Mathematics Subject Classifications
(2000): 53C43, 55R10, 58E20, 58J65, 60H30.

Keywords: harmonic sections; fiber bundles; Liouville theorem, stochastic analisys on manifolds.

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October 23, 2009

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