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We study isometric immersions f : M n −→ R n+1 into Euclidean space of dimension n + 1 of a complete Riemannian manifold of dimension n on which a compact connected group of intrinsic isometries acts with principal orbits of codimension one. We give a complete classification if either n ≥ 3 and M n is compact or if n ≥ 5 and the connected components of the flat part of M n are bounded. We also provide several sufficient conditions for f to be a hypersurface of revolution. (2000) . 53A07, 53C42.doi:10.4171/cmh/59 fatcat:kqlz5ogvbnda7amc45jgtl454m