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Geometric properties of imbedded manifolds. (Russian) Zbl 0567.53005

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

MSC:

53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
53B20 Local Riemannian geometry
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