Hypersurfaces of cohomogeneity one and hypersurfaces of revolution

Cohomogeneity one hypersurfaces of Euclidean Spaces

“…In the former case, for each principal orbit q ⊂ it follows from the Gauss equation of the restriction f | q : q → R n+1 that A ξ must be a multiple of the identity tensor, that is, the principal orbits in are umbilical in M n . This is in contradiction with Lemma 2.8 of [9], taking into account that n ≥ 5 and that f has type number 2 on .…”

“…Notice that in the latter case M n is flat outside a compact subset. We also point out that, since G is assumed to be compact, our assumption on the flat part of M n is equivalent to M n being unflat at infinity in the sense of [9].…”

“…Theorem 1.2 generalizes and gives new (and shorter) proofs of various known results. Namely, it was proved under condition (iii) in [12] in the compact case for n ≥ 4 and later in [9] in the general case (even for n = 3, 4). It was also proved in [4] (resp., [2]) in the compact case for n ≥ 5 (resp., n ≥ 4) under the assumption that all orbits have positive (resp., constant) sectional curvature.…”

Conformally flat Lorentzian hypersurfaces and curved flats

“…With another former student, J. A. Seixas, Mercuri studied in [28], the case of complete hypersurfaces. It turns out that Podesta-Spiro's theorem does not hold in this case.…”