A Note on Surfaces in Space Forms with Pythagorean Fundamental Forms

Abstract

In the present note we introduce a Pythagorean-like formula for surfaces immersed into 3-dimensional space forms M 3 (c) of constant sectional curvature c = −1, 0, 1. More precisely, we consider a surface immersed into M 3 (c) satisfying I 2 + II 2 = III 2 , where I, II and III are the matrices corresponding to the first, second and third fundamental forms of the surface, respectively. We prove that such a surface is a totally umbilical round sphere with Gauss curvature ϕ + c, where ϕ is the Golden ratio.

An Analytical Method for Tensor Visualization in a Plane

“…These references to the golden section ratio are not limited to the plane only; they can also be found in attempts to describe higher dimensions. A paper dealing with transformations of a Pythagoreanlike formula for surfaces immersed in three-dimensional space forming constant sectional curvature in a Riemann sphere examined the first, second, and third fundamental forms of the surface, proving that the immersed surfaces are totally round spheres with Gauss-like curvature ϕ + c, where ϕ is the golden ratio [28]. This, too, is a fundamental principle.…”

Pythagorean Isoparametric Hypersurfaces in Riemannian and Lorentzian Space Forms

“…We end the section by highlighting that the original definition of Pythagorean-like formula (see [6]) uses the basis of tangent vectors to the coordinate curves of given surface and in such a case the matrix A is not necessary to be diagonal. In order to better adapt to the study of isoparametric hypersurfaces in higher dimensions, we will get the basis of principal directions to use (1).…”

“…Most recently, in [6], the first and second authors gave a geometrical meaning to the matrix Pythagorean triples by using the differential geometry of surfaces.…”

“…was considered in [6], where I 2 , II 2 and III 2 are the squares of the matrices corresponding to the first, second and third fundamental forms of M 2 , respectively. We point out that {I, II, III} is a matrix Pythagorean triple and hereinafter we briefly call such a surface Pythagorean.…”

“…the hyperbolic space H 3 (−1), Euclidean space R 3 and 3-sphere S 3 (1). In [6], the authors classified compact Pythagorean surfaces immersed into M 3 (c), obtaining that a compact Pythagorean surface is a totally umbilical round sphere with Gaussian curvature G = ϕ + c.…”