Biharmonic submanifolds of pseudo-Riemannian manifolds

Abstract

In this paper, we derived biharmonic equations for pseudo-Riemannian submanifolds of pseudo-Riemannian manifolds which includes the biharmonic equations for submanifolds of Riemannian manifolds as a special case. As applications, we proved that a pseudo-umbilical biharmonic pseudo-Riemannian submanifold of a pseudo-Riemannian manifold has constant mean curvature, we completed the classifications of biharmonic pseudo-Riemannian hypersurfaces with at most two distinct principal curvatures, which were used to give four construction methods to produce proper biharmonic pseudoRiemannian submanifolds from minimal submanifolds. We also made some comparison study between biharmonic hypersurfaces of Riemannian space forms and the space-like biharmonic hypersurfaces of pseudo-Riemannian space forms. Biharmonic maps between pseudo-Riemannian manifoldsAll manifolds, maps, tensor fields are assumed to be smooth in this paper. 1.1 Some basic concepts and notations from pseudo-Riemannian geometry Let (M m t , g) be a pseudo-Riemannian (or semi-Riemannian) manifold of dimension m with a nondegenerate metric with index t. Here, nondegeneracy means the only vector X ∈ T x M satisfying g x (X, Y ) = 0 for all Y ∈ T x M and all x ∈ M is X = 0. We use |X| = |g(X, X)| 1/2 to denote the norm of a vector X and Date: 12/02/2015. 1991 Mathematics Subject Classification. 58E20, 53C12.

Biharmonic Hypersurfaces in Pseudo-Riemannian Space Forms with at Most Two Distinct Principal Curvatures

“…With the assumption that the shape operator is diagonalizable, the result of Theorem 1 was proved in [6,8,12] when the hypersurface M n r has at most three distinct principal curvatures.…”

“…Calculate Under the assumption that the shape operator is diagonalizable, the results of Theorems 6, 7, and 8 were proven not only for odd-dimensional hypersurfaces but also for even-dimensional hypersurfaces (cf. [6,8]).…”

“…Sasahara in [11] considered biharmonic hypersurfaces M 2 r of S 3 1 ðcÞ and proved that it must be minimal when r = 0, but may not when r = 1. Investigators studied biharmonic hypersurfaces with at most two distinct principal curvatures in S n+1 s ðcÞ whose shape operator is diagonalizable in [6,8] and showed that such hypersurface M n s−1 is minimal, but the hypersurface M n s may not. Naturally, there is a question as to whether any biharmonic hypersurface M n s−1 in de Sitter space S n+1 s ðcÞ is minimal.…”

“…For biharmonic hypersurface M n r with at most two distinct principal curvatures in ℍ n+1 s ðcÞ whose shape operator is diagonalizable, it is minimal when r = s and may not when r = s − 1 (cf. [6,8]). A natural question is whether any biharmonic hypersurface M n s in anti-de Sitter space ℍ n+1 s ðcÞ is minimal.…”

Biharmonic pseudo-Riemannian submersions from 3-manifolds

“…where R B is the curvature tensor of B n (see [8], [12], [18]). A smooth map ϕ is a biharmonic map (or 2-harmonic map) if its bitension field vanishes (see [12], [18]).…”