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On some special vector fields. (English) Zbl 1095.53016

The author introduces \(F\)-distinguished vector fields in a deformation algebra, where the symbol \(F\) means a \((1,1)\) tensor field. As the author writes the aim of this paper is to study these special vector fields and using their properties to characterize spherical hypersurfaces when \(F\) is the shape operator, i.e. If \(M^n\subset E^{n+1}\) is a hypersurface and \(F\) the shape operator of \(M\), then two following conditions are equivalent: i) All the elements of the algebra \(U(M,A)\) are \(F\)-distinguished vector fields. ii) \(M\) is a spherical hypersurface.
There was shown too the relation between the geometrical properties of Weyl manifolds and the algebraic properties of the Weyl algebra. These relations were introduced for: Weyl conformal connection, Levi-Civita connection and the Ricci tensor field associated to these connections.

MSC:

53B25 Local submanifolds
53B05 Linear and affine connections
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