2017/07/21 by Stylianos Stamatakis, Stamatakis, Stylianos, Ioannis Kaffas +1 · 1 citation
Mathematics · Physics and Astronomy · #53A05 #53A15 #53A40 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.1707.07549
openalex publication_date 2017/07/21 · openalex created_date 2017/11/10 · openalex updated_date 2026/07/28
We deal with hypersurfaces in the framework of the relative differential geometry in ℝ4. We consider a hypersurface \varPhi in ℝ4 with position vector field \vectx which is relatively normalized by a relative normalization \vecty. Then \vecty is also a relative normalization of every member of the one-parameter family F of hypersurfaces \varPhiμ with position vector field \vectxμ= \vectx + μ \vecty, where μ is a real constant. We call every hypersurface \varPhiμ∈ 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 \varPhiμ relatively parallel to \varPhi by means of the ones of \varPhi 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 \varPhi are constant, then there exists at least one relatively parallel hypersurface with a constant relative mean curvature's function.