It is still an open question whether a compact embedded hypersurface in theEuclidean space R^{n+1} with constant mean curvature and spherical boundary isnecessarily a hyperplanar ball or a spherical cap, even in the simplest case ofsurfaces in R^3. In a recent paper the first and third authors have shown thatthis is true for the case of hypersurfaces in R^{n+1} with constant scalarcurvature, and more generally, hypersurfaces with constant higher order r-meancurvature, when r>1. In this paper we deal with some aspects of the classicalproblem above, by considering it in a more general context. Specifically, ourstarting general ambient space is an orientable Riemannian manifold, where wewill consider a general geometric configuration consisting of an immersedhypersurface with boundary on an oriented hypersurface P. For such a geometricconfiguration, we study the relationship between the geometry of thehypersurface along its boundary and the geometry of its boundary as ahypersurface of P, as well as the geometry of P. Our approach allows us toderive, among others, interesting results for the case where the ambient spacehas constant curvature. In particular, we are able to extend the previoussymmetry results to the case of hypersurfaces with constant higher order r-meancurvature in the hyperbolic space and in the sphere.
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