Extension of two Bonnet’s theorems to the 3-dimensional relative differential geometry

Stamatakis, Stylianos; Kaffas, Ioannis; Delivos, Ioannis

doi:10.1007/s00022-017-0395-x

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Cited by 2 publications

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“…where ∇ III denotes the first Beltrami-operator with respect to the third fundamental form III of Φ (see [4, p. 197], [8]).…”

Section: Preliminariesmentioning

confidence: 99%

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

Stamatakis,

Kaffas

2017

Preprint

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We deal with hypersurfaces in the framework of the relative differential geometry in R 4 . We consider a hypersurface Φ in R 4 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 Φ µ with position vector field x µ = x + µ y, where µ is a real constant. We call every hypersurface Φ µ ∈ F relatively parallel to Φ. 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 Φ µ relatively parallel to Φ by means of the ones of Φ 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 Φ are constant, then there exists at least one relatively parallel hypersurface with a constant relative mean curvature's function.

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“…where ∇ III denotes the first Beltrami-operator with respect to the third fundamental form III of Φ (see [4, p. 197], [8]).…”

Section: Preliminariesmentioning

confidence: 99%

“…In a previous paper (see [8]) the authors and I. Delivos extended this Theorem to the relative differential geometry in R 3 in the following way:…”

Section: Introductionmentioning

confidence: 97%

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

Stamatakis,

Kaffas

2017

Preprint

Self Cite

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Equiaffine Isoparametric Functions and Their Regular Level Hypersurfaces

Hao

2019

Results Math

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Extension of two Bonnet’s theorems to the 3-dimensional relative differential geometry

Cited by 2 publications

References 1 publication

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

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

Equiaffine Isoparametric Functions and Their Regular Level Hypersurfaces

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