Indefinite Einstein hypersurfaces with nilpotent shape operators
Let \(f:M_ n^{2n}\to N^{2n+1}(c)\) be an isometric immersion of an Einstein hypersurface in an indefinite space form \(N^{(2n+1)}(c)\) (c denoting a non zero constant), and let A be the shape operator associated with f. In a previous paper [Proc. Am. Math. Soc. 84, 237-242 (1982; Zbl 0485.53023)], the author proved that if A is not diagonalizable, then there are two possibilities: either \(A^ 2=0\) or \(A^ 2=-b^ 2I\) (b denoting a non zero constant). In the present paper those \(M_ n^{2n}\) with \(A^ 2=0\) and maximal rank of A are classified. We shall quote here the following two interesting results: (i) If \(A^ 2=0\) and rank A\(=n\), then Ker(A) is an integrable totally isotropic (i.e. Ker(A)\(\subset orth.Ker(A))\) and parallel n-dimensional distribution on \(M_ n^{2n} (M_ n^{2n}\) is of signature (n,n)). (ii) Let \(f:M_ n^{2n}\to R_ n^{2n+1}\) be an isometric immersion with rank A\(=n\). Then Ker(A) is an integrable, totally isotropic, parallel distribution on \(M_ n^{2n}\), iff \(A^ 2=0\).
scientific article; zbMATH DE number 3859935
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