4-webs on hypersurfaces of 4-axial space

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.