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 Vn−1 embedded into a projective-metric space Kn, 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 Vn−1 ⊂ Kn (a tangential hypersurface \( \bar V_{n - 1} \) ⊂ \( \bar K_n \)) by quasitensor fields Hni, Hi (\( \bar H_n^i \), \( \bar H_i \)) induces a Riemannian space of constant curvature. If the two conditions are fulfilled simultaneously, the spaces Rn−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.