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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.