Sphere theorem for generalized Weingarten hypersurfaces embedded in Euclidean space

One of the most important results in differential geometry is that the only closed hypersurfaces of constant mean curvature and in general constant higher order mean curvature) embedded in Euclidean space are round spheres [1]. This result is not true for the case of immersed (and non-embedded hypersurfaces [11, 14]. Many generalizations of this result have been obtained later, for example constant scalar curvatures or constant higher order mean curvatures hypersurfaces [2,3,7,9]. As a natural generalization of hypersurfaces with constant mean curvature or with constant higher order mean curvature, linear Weingarten and more general generalized, Weingarten hypersurfaces hypersurface has been studied in many places. [5],[10]. The aim of our work is to establish a characterization theorem concerning complete generalized Weingarten hypersurfaces embedded in Euclidean space. That is an hypersurfaces where some of the higher order mean curvature are linearly related. We prove that the only closed, oriented generalized Weingarten hypersurfaces embedded in Euclidean space with non-vanishing higher order mean curvature are the round spheres. This result generalizes the cases of constant higher order mean curvature hypersurfaces and linear Weingarten hypersurfaces embedded in Euclidean space.