Shefel’, S. Z. Geometric properties of imbedded manifolds. (Russian) Zbl 0567.53005 Sib. Mat. Zh. 26, No. 1(149), 170-188 (1985). The initial moment of the present paper is the notion of a geometric property \(\Gamma\) of an n-dimensional surface in Euclidean space R, dim R\(=m\), as the invariant of some group G of transformations containing the homotheties of R (but not only them). In connection with this interpretation there arises the following scheme: 1) \(F^ n\) is a class of surfaces in R possessing the property \(\Gamma\) invariant with respect to G, and K is a class of Riemannian metrics connected by means of G by the relations. 2) Every metric of the class K admits the imbedding in R as a surface of the class \(F^ n.\) This scheme is a strong instrument to single out various classes of surfaces, both the known (for \(n=2)\) and the new ones (among them k- saddle, k-convex, k-developable) and the basis classes of metrics. For \(n=2\) the obtained classes exhaust all possible cases. Essential for \(n\geq 3\) are the following problems discussed: 1) What place, from the intrinsic viewpoint is occupied by k-saddle, k-convex and k-developable surfaces ? 2) What place is occupied by the indicated classes of metrics in the general geometry of Riemannian spaces ? Reviewer: B.N.Apanasov Cited in 1 Review MSC: 53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces 53B20 Local Riemannian geometry Keywords:k-saddle surfaces; k-convex surfaces; geometric property; k-developable surfaces PDFBibTeX XMLCite \textit{S. Z. Shefel'}, Sib. Mat. Zh. 26, No. 1(149), 170--188 (1985; Zbl 0567.53005) Full Text: EuDML