Internal geometry of hypersurfaces in projectively metric space

Abstract

In this paper, we study the internal geometry of a hypersurface V _n−1 embedded in a projectively metric space K _n , n ≥ 3, and equipped with fields of geometric-objects $\left\{ {G_n^i,{G_i}} \right\}$ and $\left\{ {H_n^i,{G_i}} \right\}$ in the sense of Norden and with a field of a geometric object $\left\{ {H_n^i,{H_n}} \right\}$ in the sense of Cartan. For example, we have proved that the projective-connection space P _n−1,n−1 induced by the equipment of the hypersurface ${V_{n - 1}}\; \subset \;{K_n},\;n \geq 3$, in the sense of Cartan with the field of a geometrical object $\left\{ {H_n^i,{H_n}} \right\}$ is flat if and only if its normalization by the field of the object $\left\{ {H_n^i,{G_i}} \right\}$ in the tangent bundle induces a Riemannian space R _n−1 of constant curvature K = −1/c.