In this paper, we give some new characterizations of umbilic hypersurfaces in general warped product manifolds, which can be viewed as generalizations of the work in KLP18 and WX14. Firstly, we prove the rigidity for hypersurfaces with constant linear combinations of shifted higher order mean curvatures. Using integral inequalities and Minkowski-type formulas, we then derive rigidity theorems in sub-static warped product manifolds, including cases that the hypersurface satisfies some nonlinear curvature conditions. Finally, we show that our results can be applied to more general warped product manifolds, including the cases with non-constant sectional curvature fiber.
Sheng et al. (Wed,) studied this question.