The authors establish that closed hypersurfaces satisfying certain constant shifted curvature equations in warped product manifolds are necessarily umbilic slices (or geodesic spheres in space forms), under conditions like star-shapedness or static-convexity.
Andrews, Contraction of convex hypersurfaces in Riemannian spaces , Journal of Differential Geometry, 1994, 39(2): 407-431
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On rigidity of hypersurfaces with constant shifted curvature functions in warped product manifolds
The authors establish that closed hypersurfaces satisfying certain constant shifted curvature equations in warped product manifolds are necessarily umbilic slices (or geodesic spheres in space forms), under conditions like star-shapedness or static-convexity.