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On Harmonic Theory in Flows
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- Journal:
- Canadian Mathematical Bulletin / Volume 46 / Issue 4 / 01 December 2003
- Published online by Cambridge University Press:
- 20 November 2018, pp. 617-631
- Print publication:
- 01 December 2003
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- Article
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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 Hörmander 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.
