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On Harmonic Theory in Flows

Published:2003-12-01
Printed: Dec 2003
• Hong Kyung Pak
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Abstract

Recently [8], a harmonic theory was developed for a compact contact manifold from the viewpoint of the transversal geometry of contact flow. A contact flow is a typical example of geodesible flow. As a natural generalization of the contact flow, the present paper develops a harmonic theory for various flows on compact manifolds. We introduce the notions of $H$-harmonic and $H^*$-harmonic spaces associated to a H\"ormander flow. We also introduce the notions of basic harmonic spaces associated to a weak basic flow. One of our main results is to show that in the special case of isometric flow these harmonic spaces are isomorphic to the cohomology spaces of certain complexes. Moreover, we find an obstruction for a geodesible flow to be isometric.
 Keywords: contact structure, geodesible flow, isometric flow, basic cohomology
 MSC Classifications: 53C20 - Global Riemannian geometry, including pinching [See also 31C12, 58B20] 57R30 - Foliations; geometric theory

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