Some properties of normal sections and geodesics on cyclic recurrent submanifolds

Let ???? be ??-dimensional (?? ? 2) submanifold in (?? + ??)-dimensional Euclidean space ????+?? (?? ? 1). Let ?? be arbitrary point ????, ???????? be tangent space to ???? at the point ??. Let ????(??, ??) be a geodesic on ???? passing through the point ?? ? ???? in the direction ?? ? ????????. Denote by ????(??, ??) and {??(??, ??) curvature and torsion of geodesic ????(??, ??) ? ????+??, respectively, calculated for point ??. Torsion {??(??, ??) of geodesic ????(??, ??) is called geodesic torsion of submanifold ???? ? ????+?? at the point ?? in the direction ??. Let ????(??, ??) be a normal section of submanifold ???? ? ????+?? at the point ?? ? ???? in the direction ?? ? ????????. Denote by ????(??, ??) and {??(??, ??) curvature and torsion of normal section ????(??, ??) ? ????+??, respectively, calculated for point ??. Denote by ?? the second fundamental form of ????, by ? the connection of van der Waerden - Bortolotti. The fundamental form ?? ?= 0 is called cyclic recurrent if on ???? there exists 1-form ?? such that ?????(??,??) = ??(??)??(??,??) + ??(?? )??(??,??) + ??(??)??(??, ?? ) for all vector fields ??, ??,?? tangent to ????. Submanifold ???? ? ????+?? with cyclic recurrent the second fundamental form ?? ?= 0 is called cyclic recurrent submanifold. The properties of normal sections ????(??, ??) and geodesics ????(??, ??) on cyclic recurrent submanifolds ???? ? ????+?? are studied in this article. The conditions for which cyclic recurrent submanifolds ???? ? ????+?? have zero geodesic torsion {??(??, ??) ? 0 at every point ?? ? ???? in every direction ?? ? ???????? are derived in this article. Denote by ?0 a set of submanifolds ???? ? ????+??, on which ????(??, ??) ?= 0, {??(??, ??) ? 0, ??? ? ????, ??? ? ????????. The following theorem is proved in this article. Let ???? be a cyclic recurrent submanifold in ????+?? with no asymptotic directions. Then ???? belongs to the set ?0 if and only if the following condition holds: ????(??, ??) = ??(??), ??? ? ????, ??? ? ????????.