Surfaces of Revolution of Frontals in the Euclidean Space
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Surfaces of Revolution of Frontals in the Euclidean Space Masatomo Takahashi1 · Keisuke Teramoto2 Received: 3 January 2019 / Accepted: 6 November 2019 © Sociedade Brasileira de Matemática 2019
Abstract For Legendre curves, we consider surfaces of revolution of frontals. The surface of revolution of a frontal can be considered as a framed base surface. We give the curvatures and basic invariants for surfaces of revolution by using the curvatures of Legendre curves. Moreover, we give properties of surfaces of revolution with singularities and cones. Keywords Surface of revolution · Frontal · Legendre curve · Framed surface Mathematics Subject Classification 57R45 · 53A05 · 58K05
1 Introduction The surface of revolution is one of classical object in differential geometry (cf. Gray et al. 2006; Kenmotsu 1980, 2003; Martins et al. 2019). It has been known that if the profile curve (the plane curve) crosses the axis of revolution, then the surface of revolution has cone-type singularity. On the other hand, if the profile curve is not regular, then the surface of revolution must have singularities. In Kenmotsu (1980, 2003), surfaces of revolution of regular curves are investigated. Especially, Kenmotsu gave concrete construction of the surface of revolution with prescribed mean curvature. In (Martins et al. 2019), the surfaces of revolution of
The first author was partially supported by JSPS KAKENHI Grant Number JP 17K05238 and the second author was partially supported by JSPS KAKENHI Grant Numbers JP 17J02151 and JP 19K14533. This work was supported by the Research Institute of Mathematical Sciences, a Joint Usage/Research Center located in Kyoto University.
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Keisuke Teramoto [email protected] Masatomo Takahashi [email protected]
1
Muroran Institute of Technology, Muroran 050-8585, Japan
2
Institute of Mathematics for Industry, Kyushu University, Motooka 744, Fukuoka 819-0395, Japan
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M. Takahashi, K. Teramoto
singular curves are investigated. They construct singular surface of revolution with prescribed (unbounded) mean curvature. In this paper, we consider more general situation. We consider frontals (Legendre curves) as singular plane curves and framed base surfaces (framed surfaces) as singular surfaces. In Fukunaga and Takahashi (2013), the authors give the curvature of Legendre curves in order to analyze Legendre curves. In Fukunaga and Takahashi (2019), the authors give the basic invariants of framed surfaces so as to analyze framed surfaces. In Sect. 2, we review the theories of Legendre curves in the unit tangent bundle over the Euclidean plane R2 and framed surfaces in the Euclidean space R3 . The surface of revolution of a frontal is a framed base surface. We can deal with surfaces of revolution with singular points more directly. In fact, we give the curvatures and basic invariants for surfaces of revolution by using the curvatures of Legendre curves in §3. Moreover, we give profile curves for given information of the curvatures, for instance, the Gauss curvature or the
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