Biharmonic Submanifolds With Parallel Mean Curvature Vector In Lorentzian Product Spaces

The study of spacelike submanifolds in pseudo-Riemannian manifolds has many important applications in problems related to Physics, more specifically in the the-ory of general relativity in recent years. Inspired by [32], this article discusses the problem of proper-biharmonic PMC submanifolds in Lorentzian product space Mn(c)×R1. Our main work includes the following aspects:First of all, by using the methods in [31], we get the Simons type equation of PMC spacelike submanifolds in product space Mn(c)×R1.Secondly, by using the methods in [11], We prove an invariant biharmonic equa-tion for biharmonic spacelike submanifolds in general pseudo-Riemannian manifolds, then we apply it to Lorentzian product manifolds Mn(c)×R1, and obtain a sufficient and necessary condition for spacelike submanifolds and apply it in the following two aspects:(i) we obtain some nonexistence results for proper-biharmonic spacelike hyper-surfaces and submanifolds in Lorentzian product manifolds Mn(c)×R1.(ii) we obtain some necessary conditions of PMC spacelike submanifolds become proper-biharmonic spacelike submanifolds in Lorentzian product manifolds Mn(c)×R1.Finally, using Simons type equation and biharmonic equation to prove a gap phenomenon for the mean curvature of a proper-biharmonic PMC submanifolds in Sn(c)×R1. In addition, we get a classification theorem of a proper-biharmonic PMC surface in Sn(c)×R1. And prove a non-pseudo-umbilical proper-biharmonic PMC spacelike surface in Sn(c)×R1is flat.