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The propagation of light and electromagnetic-gravity field in n Y space is studied. Our aims are to obtain the equations that described gravitational field in vacuum from the principle of least action in n Y space, and derive the law of motion of free point particles in n Y ; we showed that free point particles move along geodesics of space. The geodesic in n Y are depended on metric and torsion and this situation are considered in several non-trivial examples of the geodesics in the three-dimensional space without gravitational force; as result, we obtained that even in space without gravity geodesics are depending on torsion of space.

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.

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