Geodesic
Conformally Flat Spaces of Codimension 2 in a Euclidean Space
Canadian journal of mathematics, Tome 25 (1973) no. 6, pp. 1170-1173

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In a previous paper [1], the authors introduced and studied the notion of special conformally flat spaces and quasi-umbilical hypersurfaces. In that paper, the authors proved that every conformally flat space of codimension one in a Euclidean space is special and, conversely, every special conformally flat space can be isometrically immersed in a Euclidean space as a quasi-umbilical hypersurface.In the present paper, the authors study the conformally flat spaces of codimension 2 in a Euclidean space. (Manifolds, mappings, functions, etc. are assumed to be sufficiently differentiate and we shall restrict ourselves only to manifolds of dimension n > 3.) 
Chen, Bang-Yen; Yano, Kentaro. Conformally Flat Spaces of Codimension 2 in a Euclidean Space. Canadian journal of mathematics, Tome 25 (1973) no. 6, pp. 1170-1173. doi: 10.4153/CJM-1973-126-6
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[1] 1. Chen, Bang-yen and Yano, Kentaro, Special conformally flat spaces and canal hyper surf ace, Tôhoku Math. J. 25 (1973), 177–184. Google Scholar

[2] 2. Yano, Kentaro and Shigeru, Ishihara, Pseudo-umbilical submanijolds of codimension 2, Kōdai Math. Sem. Rep. 21 (1969), 365–382. Google Scholar

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