In this thesis we consider properties of total absolute geodesic curvature functional ? on circle immersions into a Riema surface based on the article [T, Ekholm, Regular Homotopy and Total curvature I: circle immersions into surfaces. (2006)]. In particular, we study the behavior of ? under regular homotopies, it is infima in regular homotopy , where ? g is the geodesic curvature of c, and where ds denotes the arc length element. We consider the simplest Riemann surface of constant curvature. First we compute the first variation of total absolute curvature . We construct similar special PGC-homotopies of curve on 2-sphere with a constant curvature metric. We show that theses PGC-homotopies can be smoothed to regular homotopies, with little increase in the total curvature. If ? is the 2-sphere then there are exactly two regular homotopy curves: one represented by a simple closed curve, the other by such a curve traversed twice. Finally, if a is a regular homotopy of curve in a Riemann surface rhen let . We allow to smooth the entire homotopy keeping control of ?. We study the space of locally convex curves on flat surface. Finally by means of obtained result, we prove the theorem of this thesis.