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A note on relatively parallel hypersurfaces in 4-dimensional relative differential geometry. (arXiv:1707.07549v3 [math.DG] UPDATED)
来源于:arXiv
We deal with hypersurfaces in the framework of the relative differential
geometry in $\mathbb{R}^4$. We consider a hypersurface $\varPhi$ in
$\mathbb{R}^4$ with position vector field $\vect{x}$ which is relatively
normalized by a relative normalization $\vect{y}$. Then $\vect{y}$ is also a
relative normalization of every member of the one-parameter family
$\mathcal{F}$ of hypersurfaces $\varPhi_\mu$ with position vector field
$\vect{x}_\mu = \vect{x} + \mu \, \vect{y}$, where $\mu$ is a real constant. We
call every hypersurface $\varPhi_\mu \in \mathcal{F}$ relatively parallel to
$\varPhi$. 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 $\varPhi_\mu$ relatively
parallel to $\varPhi$ by means of the ones of $\varPhi$ and the "relative
distance" $\mu$. Then we prove several Bonnet's type theorems. More precisely,
we show that if two relative mean 查看全文>>