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On a linearizability condition for a three-web on a two-dimensional manifold. (English) Zbl 0689.53008

Differential geometry, Proc. 3rd Int. Symp., Peñiscola/Spain 1988, Lect. Notes Math. 1410, 223-239 (1989).
[For the entire collection see Zbl 0678.00016.]
The problem of linearizability of a three-web on a two-dimensional manifold \(M^ 2\), which is directly connected with the problem of general anamorphesis in nomography [see W. Blaschke, Einführung in die Geometrie der Waben (Basel, Stuttgart 1955, Zbl 0068.365), is examined. A set of torsion-free affine connections on \(M^ 2\) for which the lines of the three-web are the geodesics is constructed and the conditions under which there exists a locally flat connection among these connections is found.
The criterion of linearizability is obtained in the form of a differential equation of third order containing the web curvature and components of the affine deformation tensor, defining the set of affine connections mentioned above. In addition to the aforementioned results the web curvature and its partial derivatives up to the third order are expressed in terms of the function \(z=f(x,y)\) defining the three-web on the (x,y)-plane. To this end a program of symbolic manipulations is used. The formulas obtained may be used for the expression of the linearizability condition of the three-web in terms of the function and its partial derivatives up to the sixth order.
Reviewer: M.A.Akivis

MSC:

53A60 Differential geometry of webs