Complex Einstein hypersurfaces of indefinite complex space forms

Abstract

In [8], Fialkow classified Einstein hypersurfaces in indefinite space forms, when the Weingarten endomorphism is diagonalizable. Recently, [13]–[15], Magid completed this work when such endomorphism is not diagonalizable. On the other hand, Smyth [17], classified complex Einstein hypersurfaces in complex space forms. Since Barros & Romero, [1], have made a systematic study of indefinite Kāhlerian manifolds, in particular, indefinite complex space forms, we investigate here complex Einstein non-degenerate hypersurfaces in indefinite complex space forms. In our case, we cannot always diagonalize the Weingarten endomorphism Aassociated to a unit normal field because it is a self-adjoint endomorphism in an indefinite metric vector space, but we prove that, if A is not diagonalizable, one has either A2 = 0 and A ≠ 0 or A2 = −γ2I, γε. We obtain in Theorems 4.4 and 5.5 a full classification except in the case that the ambient space is flat and the Weingarten endomorphism A satisfies A2 = 0and A ≠0. Our classification is based on extending Smyth's results to show that, except in this last case, an Einstein hypersurface is locally symmetric. Using expressions for the curvature tensor to estimate the dimension of the isotropy group, together with Berger's list of symmetric spaces, [3], we then determine the possible Einstein hypersurfaces.