Generalized Bishop frames of regular curves in \(\mathbb{E}^4 \). (English) Zbl 1536.53012
The geometry of a regular curve with arc-length parametrization in the Euclidean 3-dimensional space has been classically studied by means of the Frenet frame, which can be understood as an orthonormal moving frame along the curve. The Frenet formulas give the derivatives of the vectors of the Frenet frame with respect to the Frenet frame itself. They can be expressed by means of a skew-symmetric matrix. R. L. Bishop [Am. Math. Mon. 82, 246–251 (1975; Zbl 0298.53001)] studied orthonormal moving frames having the tangent vector of the curve as the first vector of the reference, and having two zeros out of the diagonal of the matrix of derivatives, finding that there are two kinds of such frames up to the change of the order of the vectors fixing the first one: the Frenet-type frames and the so-called Bishop frames (or rotation minimizing frames), which are uniquely determined up to a rotation, and can be defined for any regular curve (unlike what happens with the Frenet frame).
The authors study orthonormal moving frames along curves in the 4-dimensional Euclidean space, having the tangent vector of the curve as the first vector of the reference. They assume that there are three zeros under the diagonal of the matrix of derivatives. Such frames are classified in four classes, which are called \(B, C, D, F\). Those of type \(B\) are Bishop frames, and any curve admits such a frame. Those of type \(F\) are closely related to the Frenet frame. The main result of the paper states that if a regular curve admits a frame of type \(F\) (resp., \(D\)) then it admits a frame of type \(D\) (resp., \(C\)). The authors give examples showing that the reverse implications are not true.
The authors study orthonormal moving frames along curves in the 4-dimensional Euclidean space, having the tangent vector of the curve as the first vector of the reference. They assume that there are three zeros under the diagonal of the matrix of derivatives. Such frames are classified in four classes, which are called \(B, C, D, F\). Those of type \(B\) are Bishop frames, and any curve admits such a frame. Those of type \(F\) are closely related to the Frenet frame. The main result of the paper states that if a regular curve admits a frame of type \(F\) (resp., \(D\)) then it admits a frame of type \(D\) (resp., \(C\)). The authors give examples showing that the reverse implications are not true.
Reviewer: Fernando Etayo Gordejuela (Santander)
MSC:
| 53A04 | Curves in Euclidean and related spaces |
| 53A45 | Differential geometric aspects in vector and tensor analysis |
| 53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
Citations:
Zbl 0298.53001References:
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