Séance Séminaire

Séminaire Gaston Darboux

Pseudo-symmetry in Deszcz sense and Generalized Sasakian space form

Let $(M,g)$, $n \geq 3$, be a semi-Riemannian manifold. We consider the endomorphisms $X \wedge_{g} Y$ and $\mathcal{R}(X,Y)$ of $(M,g)$ defined by \begin{eqnarray*} X \wedge_{g} Y &=& g(Y,Z)X - g(X,Z)Y\, ,\\ \mathcal{R}(X,Y)Z &=& \nabla_{X}\nabla_{Y}Z - \nabla_{Y}\nabla_{X}Z -\nabla_{[X,Y]}Z\, , \end{eqnarray*} where $\nabla$ is the Levi-Civita connection, $\kappa$ the scalar curvature and $\mathcal{S}$ the Ricci operator of $(M,g)$. The Ricci tensor $S$ and the Ricci operator $\mathcal{S}$ of $(M,g)$ are related by $S(X,Y) = g(\mathcal{S} X, Y)$. The Riemann curvature tensor $R$ and the tensor $G$ of $(M,g)$ are defined by \begin{eqnarray*} R(X_{1},X_{2},X_{3},X_{4}) &=& g(\mathcal{R}(X_{1},X_{2})X_{3},X_{4})\, ,\\ G(X_{1},X_{2},X_{3},X_{4}) &=& g((X_{1} \wedge_{g} X_{2})X_{3},X_{4})\, , \end{eqnarray*} respectively. For a $(0,k)$-tensor $T$, $k\geq 1$, on $M$ we define the $(0,k+2)$-tensors $R\cdot T$ and $Q(g,T)$ by \begin{eqnarray*} & &(R \cdot T)(X_{1},X_{2}, \ldots , X_{k};X,Y) = (\mathcal{R}(X,Y) \cdot T)(X_{1},X_{2}, \ldots , X_{k})=\\ & &- T(\mathcal{R}(X,Y)X_{1},X_{2}, \ldots , X_{k}) - \cdots - T(X_{1},X_{2}, \ldots , \mathcal{R}(X,Y)X_{k})\, , \end{eqnarray*} \begin{eqnarray*} & &Q(g,T)(X_{1},X_{2},\ldots , X_{k};X,Y) \ =\ ((X\wedge_{g} Y) \cdot T)(X_{1},X_{2},\ldots ,X_{k})=\\ &-& T((X\wedge Y) X_{1}, X_{2}, \ldots, X_{k}) - \cdots - T(X_{1},X_{2},\ldots , X_{k-1},(X \wedge_{g} Y)X_{k}). \end{eqnarray*} If we set in the above formulas $T = R$ then we obtain the tensors: $R\cdot R$ and $Q(g,R)$. A semi-Riemannian manifold $(M,g)$, $n\geq 3$, is called semi- symmetric if $R \cdot R = 0$ on $M$. A semi-Riemannian manifold $(M,g)$, $n \geq 3$, is said to be pseudo-symmetric, if at every point of $M$ the tensors $R\cdot R$ and $Q(g,R)$ are linearly dependent. Thus we see that $(M,g)$ is pseudo-symmetric if and only if \begin{equation} R\cdot R = L_{R}\, Q(g,R) \label{pseudo} \end{equation} on $U_{R} = \{x \in M\, |\, R - \frac{\kappa }{n(n-1)}G \neq 0\ \mbox{at}\ x\}$, where $L_{R}$ is some function on $U_{R}$. In this talk, we investigate the pseudo-symmetric in Deszcz sense, of the Generalized Sasakian space from. We study also the action of $\phi$ on the curvature tensor i.e. ($\phi\cdot R$), and also conversely i.e. ($ R\cdot\phi)$,