In the plane, fixed two points, the curves that minimize the length are straight lines, i.e curves with zero curvature. In the plane, fixed an area, the curves that minimizes length are circles, i.e. curves with constant non-zero curvature. On a surface, fixed two points, the curves that minimize the length are geodesics, i.e curves with zero geodetic curvature. My question is
On a surface, fixed an area, are the curves that minimize length the ones with constant non-zero geodesic curvature?