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.
Alexsandorov, Uniqueness theorem for surfaces in the large I , Vestnik Leningrad Univ., 1956, 11: 5-17
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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.