Motivated by the quasi-local mass problem in general relativity, we study the uniqueness of isometric embedding (rigidity) into a rotationally symmetric warped product space.For the case of non space form, there is no such uniqueness even for convex closed surface, some extra conditions are needed to recover the rigidity. In this talk, the rigidity of star-shaped hypersurfaces in a spatial Schwarzschild or AdS-Schwarzschild manifold is discussed if the mean curvature doesn’t change in the isometric deformation. In another direction, we will investigate the uniqueness of isometric embedding with the geometric center fixed at the origin and with free boundary. 

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