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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 Ric Z) the tensor field of type (1,1) defined by (Y Ric Z)(X)=Ric(Z,X)Y-Ric(Y,X)Z· The authors consider the hypersurfaces M in a pseudo-Riemannian space of constant curvature c0, for which there exists a smooth real valued function f so that for any choice of vector fields X 1 ,,X 4 , Y, Z, the following relation is satisfied at any xM:

(R YZ ·W)(X 1 ,,X 4 ) x =f(x)·W((Y Ric Z)(X 1 ),,X 4 )++
+W(X 1 ,,(Y Ric Z)(X 4 ))·

They prove that any such hypersurface is of type number two and satisfies certain additional restriction on R, Ric and W.

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