On the energy of sections of parallelizable sphere bundles

SALVAI, MARCOS

RENDICONTI DEL SEMINARIO MATEMATICO (TORINO)

Rendiconti del Seminario Matematico

Año: 2002 vol. 60 p. 147 - 155

0373-1243

Let E -> M be a vector bundle with a metric connection over a Riemannian manifold M and consider on E the Sasaki metric. We find a condition for a section of the associated sphere bundle to be a critical point of the energy among all smooth unit sections. We apply the criterion to some particular cases where M is parallelizable, for instance M = S7 or a compact simple Lie group G with a bi- invariant metric, and E is the trivial vector bundle with a connection induced by octonian multiplication or an irreducible real orthogonal representation of G; respectively. Generically, these bundles have no parallel unit sections.