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Differential Geometry of Manifolds

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Kretov M. V.

Differentiable mapping generated by elliptic parabo­loid complexes

DOI
10.5922/0321-4796-2020-51-9
Pages
76-80
complex equiaffine space mapping elliptical paraboloid main direction display index system of Pfaff equations coordinate vec­tors reference characteristic variety

Abstract

In three-dimensional equiaffine space, we consider a differentiable map generated by complexes with three-parameter families of elliptic paraboloids according to the method proposed by the author in the mate­rials of the international scientific conference on geometry and applica­tions in Bulgaria in 1986, as well as in works published earlier in the sci­entific collection of Differ. Geom. Mnogoobr. Figur. The study is carried out in the canonical frame, the vertex of which coincides with the top of the generating element of the manifold, the first two coordinate vectors are conjugate and lie in the tangent plane of the elliptic paraboloid at its vertex, the third coordinate vector is directed along the main diameter of the generating element so that the ends are, respectively, the sums of the first and third, and also the sums of the second and third coordinate vec­tors lay on a paraboloid, while the indicatrixes of all three coordinate vec­tors describe lines with tangents, parallel to the first coordinate vector. The existence theorem of the mapping under study is proved, according to which it exists and is determined with the arbitrariness of one function of one argument. The systems of equations of the indicatrix and the main directions of the mapping under consideration are obtained. The indicatrix and the cone of the main directions of the indicated mapping are geomet­rically characterized.

Reference

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