Axial Curvature Cycles of Surfaces Immersed in R4

In this paper are established integral expressions, in terms of geometric invariants along a closed curve-a cycle-of axial curvature, which characterize hyperbolicity i.e. non unitiy of the derivative for the Poincare first return map-holonomy-at the axial curvatue cycle on a regular surface M immersed in R-4. A proof of the genericity of hyperbolicity is given here. An integral expression for the second derivative in terms of higher order geometric invariants along a non-hyperbolic axial curvature cycle is also established in this paper. This work improves results obtained by the first and second authors.