Dual Riemannian spaces of constant curvature on a normalized hypersurface

In this paper we obtain the following results: 1) we prove that in a differential neighborhood of the fourth order a regular hypersurface $\mathrm V_{n-1}$ embedded in a projective-metric space $\mathrm K_n$, $n\geqslant3$, intrinsically induces the dual projective-metric space $\overline K_n$; 2) we obtain an invariant analytical condition under which the normalization of the hypersurface $\mathrm V_{n-1}\subset\mathrm K_n$ (the tangential hypersurface $\overline{\mathrm V}_{n-1}\subset\overline{\mathrm K}_n$) by fields of quasitensors $H^i_n$, $H_i$ ($\overline H^i_n$, $\overline H_i$) induces a Riemannian space of constant curvature. Note that when these two conditions are fulfilled simultaneously, spaces $R_{n-1}$ and $\overline R_{n-1}$ are dual with the same identical constant curvature $\mathrm K=-\frac1c$; 3) we give geometric descriptions of the obtained analytical conditions.