Theory of Hypersurfaces yn–1 in yn Space and Geodesics on this Hypersurfaces yn–1
Yaremenko Mikola (Nikolay) Ivanovich *
Department of Advance Mathematics, National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", Kyiv 03056, Solomenskiy District, Ave. Pobedy, 37, Ukraine and Institute of Mathematics of the National Academy of Sciences of Ukraine, Kiev 010044, Tereschenkivska St. 3, UkraineDepartment of Mathematics, Claremont Graduate University, USA.
*Author to whom correspondence should be addressed.
Abstract
The hypersurface yn–1 in yn space is studied for this piece of work. We established the correlation between tensors of hypersurface yn–1 and tensors of embedding space yn . The second non-symmetrical tensor of hypersurface has been introduced, which have been obtained from the analog of Peterson-Codazzi equation in nonsymmetrical case.Also we have introduced the tensor that is associated with square of angle between normal and adjacent normal and it is represented in terms of metric and second tensors of hypersurface. The geodesics on hypersurface have been studied, and nontrivial example of geodesics on hypersurface with torsion and Euclid metric was constructed.
Keywords: Hypersurface, gravity, geometry, geodesic, torsion, second tensor of hypersurface, connection