Abstract
LeM be a (2m+2)-dimensional Riemannian manifold with two structure vector fieldsξ r (r=2m+1, 2m+2) and letη r=ξ b r be their corresponding covectors (or Pfaffians). These vector fields define onM a 2-almost contact structure.
If the 2-formϕ=η 2m+1∧η 2m+2 is harmonic, then, following S. Tachibana [12],M is a Tachibana manifold and in this caseM is covered with 2 families of minimal submanifolds tangent toD ⊥={ξ r} and its complementary orthogonal distributionD ⊤.
On such a manifold a canonical eigenfunction α of the Laplacian is associated. Since the corresponding eingenvalue is negative,M cannot be compact. Any horizontal vector fieldX orthogonal to α# is a skew-symmetric Killing vector field (see [6]).
Next, we assume that the Tachibana manifoldM under consideration is endowed with a framedf-structure defined by an endomorphism ϕ of the tangent bundleTM. Infinitesimal automorphisms of the symplectic form Ω ϕ are obtained.
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Mihai, I., Rosca, R. & Verstraelen, L. On 2-almost contact tachibana manifolds. Rend. Circ. Mat. Palermo 50, 415–426 (2001). https://doi.org/10.1007/BF02844422
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DOI: https://doi.org/10.1007/BF02844422