Internal geometry of hypersurfaces in projectively metric space

A. V. Stolyarov

Fundamentalʹnaâ i prikladnaâ matematika, Tome 16 (2010) no. 2, pp. 103-114

Résumé

In this paper, we study the internal geometry of a hypersurface $\mathrm V_{n-1}$ embedded in a projectively metric space $\mathrm K_n$, $n\ge3$, and equipped with fields of geometric-objects $\{G^i_n,G_i\}$ and $\{H^i_n,G_i\}$ in the sense of Norden and with a field of a geometric object $\{H^i_n,H_n\}$ in the sense of Cartan. For example, we have proved that the projective-connection space $\mathrm P_{n-1, n-1}$ induced by the equipment of the hypersurface $\mathrm V_{n-1}\subset\mathrm K_n$, $n\ge3$, in the sense of Cartan with the field of a geometrical object $\{H^i_n,H_n\}$ is flat if and only if its normalization by the field of the object $\{H^i_n,G_i\}$ in the tangent bundle induces a Riemannian space $R_{n-1}$ of constant curvature $\mathrm K=-1/c$.