FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2010, VOLUME 16, NUMBER 2, PAGES 103-114

Internal geometry of hypersurfaces in projectively metric space

A. V. Stolyarov

Abstract

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In this paper, we study the internal geometry of a hypersurface $V$_n-1 embedded in a projectively metric space $K$_n, $n$³ 3, and equipped with fields of geometric-objects $\{Gi$_n,G_i} and $\{Hi$_n,G_i} in the sense of Norden and with a field of a geometric object $\{Hi$_n,H_n} 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 Ì K_n, $n$³ 3, in the sense of Cartan with the field of a geometrical object $\{Hi$_n,H_n} is flat if and only if its normalization by the field of the object $\{Hi$_n,G_i} in the tangent bundle induces a Riemannian space $R$_n-1 of constant curvature $K\; =$- 1/c.

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Last modified: April 5, 2011