Existence and uniqueness for Legendre curves

We give a moving frame of a Legendre curve (or, a frontal) in the unit tangent bundle and define a pair of smooth functions of a Legendre curve like as the curvature of a regular plane curve. It is quite useful to analyse the Legendre curves. The existence and uniqueness for Legendre curves hold similarly to the case of regular plane curves. As an application, we consider contact between Legendre curves and the arc-length parameter of Legendre immersions in the unit tangent bundle.