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Biharmonic pseudo-Riemannian submanifolds in pseudo-Euclidean spaces. (English) Zbl 0892.53012

Let x:M n E s m be an isometric immersion of a pseudo-Riemannian n-manifold M into the pseudo-Euclidean space E s m of signature s. The immersion is called biharmonic if the immersion vector x satisfies the equation Δ 2 x=0, where Δ is the Laplace operator on M. The first author of this paper conjectured that the only biharmonic submanifolds of Euclidean spaces are the minimal ones. The conjecture was confirmed for some classes of submanifolds. In contrast to the Euclidean case, there are non-minimal and biharmonic submanifolds in pseudo-Euclidean spaces.

In the paper under review, the authors consider pseudo-Riemannian biharmonic submanifolds in the pseudo-Euclidean space E s m , s=1,2,3, and give several classifications.

53B25Local submanifolds
53B30Lorentz metrics, indefinite metrics