Bonnet's type theorems in the relative differential geometry of the 4-dimensional space

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We deal with hypersurfaces in the framework of the relative differential geometry in R⁴. We consider a hypersurface in R⁴ with position vector field x which is relatively normalized by a relative normalization y. Then y is also a relative normalization of every member of the one-parameter family F of hypersurfaces μ with position vector field xμ= x + μ y, where μ is a real constant. We call every hypersurface μ∈ F relatively parallel to. This consideration includes both Euclidean and Blaschke hypersurfaces of the affine differential geometry. In this paper we express the relative mean curvature's functions of a hypersurface μ relatively parallel to by means of the ones of and the "relative distance" μ. Then we prove several Bonnet's type theorems. More precisely, we show that if two relative mean curvature's functions of are constant, then there exists at least one relatively parallel hypersurface with a constant relative mean curvature's function.