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Gradient vector fields on cosymplectic manifolds. (English) Zbl 0754.53024

Let \(M(\omega,\theta)\) be a \((2n+1)\)-dimensional cosymplectic \(C^ \infty\)-manifold, that is the structure 2-form \(\omega\) and the structure 1-form \(\theta\) satisfy \(d\omega=0\), \(d\theta=0\), \(\theta\wedge\omega^ n\neq 0\). Let \(\chi_{\theta,\omega}: TM\to T^*M\) be the bundle homomorphism defined by \((\omega,\theta)\) and \(\Omega_ M=-d\Theta_ M\), be the canonical symplectic form on \(T^*M\) where \(\Theta_ M\) denotes the canonical (or Liouville) 1-form. By pulling back \(\Omega_ M\) to \(TM\), let \(\Omega_ 0=\chi_{\theta\omega}^*\Omega_ M\) be the symplectic form on \(TM\). Making use of some of the above concepts the authors study different properties of gradient vector fields on \(M(\omega,\theta)\). We quote here the simplest: A vector field \(X\) on a cosymplectic manifold \(M(\omega,\theta)\) is a local gradient vector field iff \(im(X)\) is a Lagrangian submanifold of \((TM,\Omega_ 0)\), (a subspace \(L\) of a symplectic vector space \((V,\Omega)\) is Lagrangian iff \(L=L^ \perp\)).
Reviewer: R.Roşca (Paris)

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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