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Natural vector fields and 2-vector fields on the tangent bundle of a pseudo-Riemannian manifold. (English) Zbl 1090.58007

Let \(M\) be a manifold with a pseudo-Riemannian metric \(g\) and a linear symmetric connection \(K\). The author classifies all natural \(0\)-order vector fields \(E\) and \(2\)-vector fields \(\Lambda \) on \(TM\) generated by \(g\) and \(K\). Conditions for \((E,\Lambda )\) to define a Jacobi or Poisson structure on \(TM\) are discussed.

MSC:

58A32 Natural bundles
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
58A20 Jets in global analysis
53D17 Poisson manifolds; Poisson groupoids and algebroids
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