Legendre curves in the unit spherical bundle over the unit sphere and evolutes

Author:
  • MTMasatomo Takahashi
    Masatomo Takahashi
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Abstract
In order to consider singular curves in the unit sphere, we consider Legendre curves in the unit spherical bundle over the unit sphere. By using a moving frame, we define the curvature of Legendre curves in the unit spherical bundle. As applications, we give a relationship among Legendre curves in the unit spherical bundle, Legendre curves in the unit tangent bundle and framed curves in the Euclidean space, respectively. Moreover, we define not only an evolute of a front, but also an evolute of a frontal in the unit sphere under certain conditions. Since the evolute of a front is also a front, we can take evolute again. On the other hand, the evolute of a frontal if exists, is also a frontal. We give an existence and uniqueness conditions of the evolute of a frontal.

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