Dual Riemannian spaces of constant curvature on a normalized hypersurface

Abstract

In this paper, the following results are obtained: 1) It is proved that, in the fourth order differential neighborhood, a regular hypersurface V_n−1 embedded into a projective-metric space K _n , n ≥ 3, intrinsically induces a dual projective-metric space $\bar K_n$. 2) An invariant analytical condition is established under which a normalization of a hypersurface V_n−1 ⊂ K _n (a tangential hypersurface $\bar V_{n - 1}$ ⊂ $\bar K_n$) by quasitensor fields H ⁱ _n , H _i ($\bar H_n^i$, $\bar H_i$) induces a Riemannian space of constant curvature. If the two conditions are fulfilled simultaneously, the spaces R _n−1 and $\bar R_{n - 1}$ are spaces of the same constant curvature $K = - \tfrac{1} {c}$. 3) Geometric interpretations of the obtained analytical conditions are given.