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On hypersurfaces with type number two in space forms. (English) Zbl 1083.53023
For any pseudo-Riemannian manifold $(M,g)$, let us denote by $R$, Ric and $W$ the Riemann tensor, the Ricci tensor and the Weyl tensor, respectively. Also, for any two vector fields $Y, Z$, let us denote by $(Y \wedge_{\text{Ric}}Z)$ the tensor field of type $(1,1)$ defined by $(Y \wedge_{\text{Ric}}Z)(X) = \text{Ric}(Z, X) Y - \text{Ric}(Y, X) Z.$ The authors consider the hypersurfaces $M$ in a pseudo-Riemannian space of constant curvature $c \neq 0$, for which there exists a smooth real valued function $f$ so that for any choice of vector fields $X_1, \dots, X_4$, $Y$, $Z$, the following relation is satisfied at any $x \in M$: $$\left.(R_{Y Z} \cdot W)(X_1, \dots, X_4)\right\vert _x = f(x)\cdot \left\{W((Y\wedge_{\text{Ric}} Z)(X_1), \dots, X^4) + \dots + \right.$$ $$ \left. + W( X_1, \dots, (Y\wedge_{\text{Ric}} Z) (X^4)) \right\}.$$ They prove that any such hypersurface is of type number two and satisfies certain additional restriction on $R$, Ric and $W$.

53B25Local submanifolds
53C42Immersions (differential geometry)
53B30Lorentz metrics, indefinite metrics