We give a moving frame of a Legendre curve (or, a frontal) in the unite tangent bundle and define a pair of smooth functions of a Legendre curve like as the curvature of a regular plane curve. The existence and uniqueness for Legendre curves are holded like as regular plane curves. It is quite useful to analyse the Legendre curves. As applications, we consider contact between Legendre curves and the arc-length parameter of Legendre immersions in the unite tangent bundle.
A framed curve in the Euclidean space is a curve with a moving frame. It is a generalization not only of regular curves with linear independent condition, but also of Legendre curves in the unit tangent bundle. We de ne smooth functions for a framed curve, called the curvature of the framed curve, similarly to the curvature of a regular curve and of a Legendre curve. Framed curves may have singularities. The curvature of the framed curve is quite useful to analyse the framed curves and their singularities. In fact, we give the existence and the uniqueness for the framed curves by using their curvature. As applications, we consider a contact between framed curves, and give a relationship between projections of framed space curves and Legendre curves.
The evolute of a regular curve in the Euclidean plane is given by not only the caustics of the regular curve, envelope of normal lines of the regular curve, but also the locus of singular loci of parallel curves. In general, the evolute of a regular curve have singularities, since such a point is corresponding to a vertex of the regular curve and there are at least four vertices for simple closed curves. If we repeated an evolute, we cannot define the evolute at a singular point. In this paper, we define an evolute of a front and give properties of such evolute by using a moving frame of a front and the curvature of the Legendre immersion. As applications, repeated evolutes can be well-defined and these are useful to recognize the shape of curves.
We consider extrinsic differential geometry on spacelike hypersurfaces in Minkowski pseudo-spheres (hyperbolic space, de Sitter space and the lightcone). In the previous paper  we have shown a basic Legendrian duality theorem between pseudo-spheres. We define the spacelike parallels by using the basic Legendrian duality theorem. This definition unifies the notions of parallels of spacelike hypersurfaces in pseudo-spheres. We also define the evolute as the locus of singularities of the spacelike parallels. These notions are investigated as applications of Lagrangian or Legendrian singularity theory. We consider geometric properties of non-singular spacelike hypersurfaces corresponding to singularities of spacelike parallels or evolutes.
Some geometrical properties associated to the contact of submanifolds with hyperhorospheres in hyperbolic n-space are studied as an application of the theory of Legendrian singularities.
. We have already defined the evolutes and the involutes of fronts without inflection points. For regular curves or fronts, we can not define the evolutes at inflection points. On the other hand, the involutes can be defined at inflection points. In this case, the involute is not a front but a frontal at inflection points. We define evolutes of frontals under conditions. T he definition is a gener alisation of both evolutes of regular curves and of fronts. By using relationship between evolutes and involutes of frontals, we give an existence condition of the evolute with inflection points. We also give properties of evolutes and involutes of frontals. IntroductionT he notions of evolutes and involutes (also known as evolvents) were studied by C. Huygens in his work  and studied in classical analysis, differential geometry and singularity theory of planar curves ( cf. [3, 4, 6, 10, 11, 12, 17]). T he evolute of a regular curve in the Euclidean plane is given by not only the locus of all its centres of the curvature (the caustics of the regular curve) , but also the envelope of normal lines of the regular curve, namely, the locus of singular loci of parallel curves (the wave front of the regular curve). On the other hand, the involute of a regular curve is to replace the taut string by a line segment that is tangent to the curve on one end, while the other end traces out the involute. T he length of the line segment is changed by an amount equal to the arc length traversed by the tangent point as it moves along the curve.In the previous papers [8,9], we defined the evolutes and the involutes of fronts without inflection points and gave properties of them. In §2, we review the evolutes and the involutes of regular curves and of fronts. We introduce (cf. ). Moreover, we also gave properties of the evolutes and the involutes of fronts, for more detail see [8,9]. For a Legendre immersion without inflection points, the evolute and the involute of the front are also fronts without inflection points. It follows that we can repeat the evolute and the involute of fronts without inflection points. We gave the n-th form of evolutes and involutes of fronts without inflection points for all n P N in [8,9]. The evolute and the involute of the front without inflection points are corresponding to the differential and the integral of the curvatures of the Legendre immersions.The evolutes of fronts can not be defined at inflection points. On the other hand, the involutes of fronts can be defined at inflection points. In this case, the involute is a frontal at inflection points. In this paper, we consider evolutes and involutes of frontals under conditions. In §3, we define evolutes and involutes of frontals by extending to the evolutes and the involutes of fronts. These definitions are generalisations of evolutes and involutes of regular curves and of fronts. Even if evolutes of frontals exist, we don't know whether evolutes of evolutes exist or not. By using relationship between evolutes and involutes of frontals...
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