We study isometric immersions into Euclidean space of dimension of a complete Riemannian manifold of dimension on which a compact connected group of intrinsic isometries acts with principal orbits of codimension one. We give a complete classification if either and is compact or if and the connected components of the flat part of are bounded. We also provide several sufficient conditions for to be a hypersurface of revolution.
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Francesco Mercuri, Fabio Podestà, José A. P. Seixas, Ruy Tojeiro, Cohomogeneity one hypersurfaces of Euclidean Spaces. Comment. Math. Helv. 81 (2006), no. 2, pp. 471–479DOI 10.4171/CMH/59