Hypertangent BundlesW(M 2n ) with (H, ξ, η, G)-structure(M 2n ) with (H, ξ, η, G)-structure

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LetJ(M2n) be a vector bundle of (2n+1) dimensions over a differentiable manifold M2n. IfJ(M2n) admits an almost contact metric structure,i.e., (Φ, ξ, η, G)-structure, then it is a hypertagent bundle over M2n and the base manifold M2n admits an almost Hermitian structure. If in such a hypertangent bundleW(M2n) with (Φ, ξ, η, G)-structure the structure tensor field coincides with the second fundamental tensor field H ofW(M2n), we call it the hypertangent bundleW(M2n) with (H, ξ, η, G) structure. The base manifold M2n of such a hypertangent bundle is always totally geodesic The Nijenhuis tensor of the induced almost complex structure which is the second fundamental tensor field of M2n can be expressed interms of the covector field η ofW(M2n) and of the Poisson bracket operator [Bc, Bb] made by the tangent frame vectors Bb of M2n. If h is equal to the covariant derivative of a vector field u in M, one calls such aW(M2n) the hypertangent bundle with\((H,\xi ,\eta ,G:h = \mathop \nabla \limits^o u)\)-structure. In this case u is a Killing vector field and is a contravariant almost analytic vector field at the same time. Its base manifold M2n admits an almost Kaehlerian structure.


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This work was supported by National Research Council, Canada, A-4037 (1967-’68).

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Okubo, T. Hypertangent BundlesW(M 2n ) with (H, ξ, η, G)-structure(M 2n ) with (H, ξ, η, G)-structure. Annali di Matematica 78, 159–185 (1968) doi:10.1007/BF02415114

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  • Vector Field
  • Vector Bundle
  • Covariant Derivative
  • Poisson Bracket
  • Tensor Field