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Some remarks on metric semisymmetric connections. (Quelques remarques sur les connexions semi-symétriques métriques.) (French) Zbl 1100.53025
Let \(M\) be a smooth manifold, \({\mathcal F}(M)\) the ring of smooth real functions on \(M\) and \({\mathcal T}_{s}^{r}(M)\) the \({\mathcal F}(M)\)-module of \((r,s)\)-tensor fields on \(M\). One sets \({\mathcal T}_{0}^{1}(M)= {\mathcal X}(M)\) and \({\mathcal T}_{1}^{0}(M)= \Lambda^{1}(M)\). For \(A\in{\mathcal T} _{2}^{1}(M)\) one defines the product \(X \circ Y =A(X,Y)\), \(X,Y\in{\mathcal X} (M)\) and so \({\mathcal X}(M)\) becomes an \({\mathcal F}(M)\)-algebra denoted by \({\mathcal U}(M,A)\). If \(\nabla\) and \(\overline{\nabla}\) are two linear connections on \(M\), the algebra \({\mathcal U}(M,\overline{\nabla}-\nabla)\) is called the deformation algebra of the pair \((\nabla,\overline{\nabla})\). Let \((M,g)\) be a Riemannian manifold. A linear connection \(\nabla\) on \(M\) is called metric semi-symmetric if \(\nabla g=0\) and the torsion field \( T(x,y)=\pi(Y)X-\pi(X)Y\) for some \(\pi\in\Lambda^{1}(M)\). The authors consider also the Levi-Civita connection \(\overset{\circ}{\nabla}\), the transpose \(\overline{\nabla}\) of \(\nabla\) and the symetric connection \(\widetilde{\nabla}\) associated to \(\nabla\) and provide several conditions for the deformation algebras \({\mathcal U}(M,\nabla -\overset{\circ}{\nabla} )\), \({\mathcal U}(M,\overline{\nabla }-\overset{\circ}{\nabla} )\), \({\mathcal U}(M,\widetilde{\nabla}-\overset{\circ}{\nabla} )\) to be associative.
Reviewer: Radu Miron (Iaşi)
MSC:
53B20 Local Riemannian geometry
53B21 Methods of local Riemannian geometry
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