On the shape operator of relatively parallel hypersurfaces in the n-dimensional relative differential geometry

2017/12/28 by Stylianos Stamatakis, Stamatakis, Stylianos, Ioannis Kaffas +1
Mathematics · #53A05 #53A15 #53A40 #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.1712.10319

openalex publication_date 2017/12/28 · openalex created_date 2018/01/05 · openalex updated_date 2026/07/28

Abstract

We deal with hypersurfaces in the framework of the n-dimensional relative differential geometry. We consider a hypersurface \varPhi of ℝⁿ⁺¹ 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 \varPhi_μ with position vector field x_μ= x + μ y, where μ is a real constant. We call every hypersurface \varPhi_μ∈ F relatively parallel to \varPhi at the "relative distance" μ. In this paper we study (a) the shape (or Weingarten) operator, (b) the relative principal curvatures, (c) the relative mean curvature functions and (d) the affine normalization of a relatively parallel hypersurface ( \varPhi_μ,y) to (\varPhi,y).

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