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Air Development Center Air Force Base Air Research ATTNs Technical Library Base Florida ATTN Base Ohio ATTNs Betti number Cartan Cartan's theorem Center Wright-Patterson Air closed orbit closed trajectories cocycles cohomology Commander Air Force compact Lie group cup product defined denote Department of Mathematics differential forms divergence free p-tensor divergence-free dt(m dTs(w exterior product FLOWS ON MANIFOLDS Force Base Ohio FRANKEL TECHNICAL NOTE G is compact G s m h.v1 dt hence homogeneous space homologous to zero HOMOLOGY AND FLOWS homology class homology group invariant under G invariant volume element isometric flow Killing vector Library 1 Commander linear space N. Y. ATTN Office of Scientific Ohio ATTNs Technical one-parameter sub one-parameter subgroup p-form point transformations positive scalar density PROOF Research and Development Rham's theorem Riemannian manifold right invariant s m s m subgroup of G tangent vector tensors USAF vector field vector formulation Wright-Patterson Air Force