On generalised framed surfaces in the Euclidean space

Masatomo Takahashi; Haiou Yu; Masatomo Takahashi; Haiou Yu

doi:10.3934/math.2024861

AIMS Mathematics

2024, Volume 9, Issue 7: 17716-17742. doi: 10.3934/math.2024861

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Research article

On generalised framed surfaces in the Euclidean space

Masatomo Takahashi ¹,
Haiou Yu ^{2
,
,}

1.
Department of Mathematical Science, Muroran Institute of Technology, Muroran 0508585, Japan
2.
Department of Mathematics, Jilin University of Finance and Economic, Changchun 130117, China

Received: 05 April 2024 Revised: 13 May 2024 Accepted: 15 May 2024 Published: 23 May 2024
MSC : 58K05, 53A05, 57R45

We have introduced framed surfaces as smooth surfaces with singular points. The framed surface is a surface with a moving frame based on the unit normal vector of the surface. Thus, the notion of framed surfaces (respectively, framed base surfaces) is locally equivalent to the notion of Legendre surfaces (respectively, frontals). A more general notion of singular surfaces, called generalised framed surfaces, is introduced in this paper. The notion of generalised framed surfaces includes not only the notion of framed surfaces, but also the notion of one-parameter families of framed curves. It also includes surfaces with corank one singularities. We investigate the properties of generalised framed surfaces.
- generalised framed surface,
- framed surface,
- parallel surface,
- singular point,
- basic invariants
Citation: Masatomo Takahashi, Haiou Yu. On generalised framed surfaces in the Euclidean space[J]. AIMS Mathematics, 2024, 9(7): 17716-17742. doi: 10.3934/math.2024861

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Abstract

We have introduced framed surfaces as smooth surfaces with singular points. The framed surface is a surface with a moving frame based on the unit normal vector of the surface. Thus, the notion of framed surfaces (respectively, framed base surfaces) is locally equivalent to the notion of Legendre surfaces (respectively, frontals). A more general notion of singular surfaces, called generalised framed surfaces, is introduced in this paper. The notion of generalised framed surfaces includes not only the notion of framed surfaces, but also the notion of one-parameter families of framed curves. It also includes surfaces with corank one singularities. We investigate the properties of generalised framed surfaces.

References

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