FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2010, VOLUME 16, NUMBER 1, PAGES 65-79

$4$-webs on hypersurfaces of $4$-axial space

V. V. Zabrodin

Abstract

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V. B. Lazareva investigated $3$-webs formed by shadow lines on a surface embedded in $3$-dimensional projective space is assuming that the lighting sources are situated on $3$ straight lines. The results were used, in particular, for the solution of Blaschke problem of classification of regular $3$-webs formed by pencils of circles in a plane. In the present paper, we consider a $4$-web $W$ formed by shadow surfaces on a hypersurface $V$ embedded in $4$-dimensional projective space assuming that the lighting sources are situated on $4$ straight lines. We call the projective $4$-space with $4$ fixed straight lines a $4$-axial space. Structure equations of $4$-axial space and of the surface $V$, asymptotic tensor of $V$, torsions and curvatures of $4$-web $W$, and connection form of invariant affine connection associated with $4$-web $W$ are found.

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Last modified: March 11, 2011