Abstract. 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. Mathematics Subject Classification (2000). 53A07, 53C42.

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In this paper, we introduce the notion of tube with arbitrary cross sections around a curve for which we calculate the volume and give a generalization for the second theorem of Pappus. The first Theorem of Pappus is obtained for sphere tubes in arbitrary dimension.