Flat double rotational surfaces in Euclidean and Lorentz-Minkowski 4-space

Abstract

A new type of surfaces in 4-dimensional Euclidean and Lorentz-Minkowski space is constructed by performing two simultaneous rotations on a planar curve. In analogy with rotational surfaces, the resulting surfaces are called double rotational surfaces. Classification theorems of flat double rotational surfaces are proved. These classifications contain amongst other cones over 4-dimensional Clelia curves. As a side product these new kinds of curves in 4-space are defined.

Twisted Surfaces in Semi-Euclidean $4$-Space with Index $2$

“…They also showed that there exists no minimal (regular) twisted surface in Euclidean or in Minkowski 3-space when excluding the surfaces of revolution. In another study by the same authors, the curvature properties of twisted surfaces with null rotation axis in Minkowski 3-space were studied [6].…”

“…Similar to the generalization of rotational surfaces and their related properties from 3-dimensional Euclidean space to 4-dimensional space [13], Twisted surfaces are defined in 4-dimensional Euclidean space and 4dimensional Lorentz space and their related properties are studied in detail [6].…”

The Clairaut's theorem on rotational surfaces in pseudo Euclidean 4-space with index 2

“…In [5], a brief description of the hyperbolic and elliptic rotational surfaces is defined using a curve and matrices in 4-dimensional semi-Euclidean space and different types of rotational matrices which are the subgroups of M by rotating a selected axis in E 4 are given by the authors. In [8], A new type of surfaces in Euclidean and Lorentz-Minkowski 4-space are constructed by performing two simultaneous rotations on a planar curve by the authors. Also, the classification theorems of flat double rotational surfaces are proved by the authors.…”

The physical approach on the surfaces of rotation in $E_{2}^{4}$

“…In [5], a brief description of the hyperbolic and elliptic rotational surfaces is defined using a curve and matrices in semi-Euclidean 4-space and different types of rotational matrices which are the subgroups of M by rotating the axis in E 4 are given by the authors. In [8], A new type of surfaces in Euclidean and Lorentz-Minkowski 4-space is constructed by performing two simultaneous rotations on a planar curve by the authors. Also, classification theorems of flat double rotational surfaces are proved by the authors.…”