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

This paper is devoted to the 3-dimensional relative differential geometry of surfaces. In the Euclidean space E 3 we consider a surface Φ with position vector field x , which is relatively normalized by a relative normalization y . A surface Φ ∗ with position vector field x ∗ = x + μ y , where μ is...

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Bibliographic Details
Published in:Journal of geometry 2017-12, Vol.108 (3), p.1073-1082
Main Authors: Stamatakis, Stylianos, Kaffas, Ioannis, Delivos, Ioannis
Format: Article
Language:English
Subjects:
Differential geometry
Euclidean geometry
Euclidean space
Geometry
Mathematics
Mathematics and Statistics
Theorems
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Summary:This paper is devoted to the 3-dimensional relative differential geometry of surfaces. In the Euclidean space E 3 we consider a surface Φ with position vector field x , which is relatively normalized by a relative normalization y . A surface Φ ∗ with position vector field x ∗ = x + μ y , where μ is a real constant, is called a relatively parallel surface to Φ . Then y is also a relative normalization of Φ ∗ . The aim of this paper is to formulate and prove the relative analogues of two well known theorems of O. Bonnet which concern the parallel surfaces (see Bonnet in Nouv Ann de Math 12:433–438, 1853 ).
ISSN:0047-2468
1420-8997
DOI:10.1007/s00022-017-0395-x