Biconservative Hypersurfaces in $$\mathbb {E}^4_1$$ with Non-diagonalizable Shape Operator

Abstract

The authors study biconservative isometric immersions into Minkowski \(4\)-space \(\mathbb{E}^4_1\). They focus on hypersurfaces with non-diagonalizable shape operator. The diagonalizable case has been studied by \textit{Y. Fu} and the third author [Int. J. Math. 27, No. 5, Article ID 1650041, 17 p. (2016; Zbl 1339.53054)]. The authors assume that \(M\) is a proper biconservative Lorentzian hypersurface with non-diagonalizable shape operator \(A\) in the Minkowski space \(\mathbb{E}^4_1\). First, they obtain the matrix representation of \(A\) with respect to a pseudo-orthonormal frame field on \(M\). Then they prove a uniqueness theorem for isometric immersions of this type of hypersurfaces into \(\mathbb{E}^4_1\) (Theorem 3.4). Finally, they give the local classification of proper biconservative hypersurfaces with non-diagonalizable shape operator (Theorem 3.12).