In this paper, we investigate the behavior of the curvature of non-developable surfaces around an umbilic point at the origin. The surfaces are of the form z = f(x,y) where f is a nonhomogeneous bivariate polynomial with cubic and quartic terms. We do this by looking at the continuity of the principal directions around the origin as well as the rate that the principal curvatures converge to zero as they approach the origin. This is done by considering the eigenvectors and eigenvalues of the shape operator. In our main result, we prove that a continuously diagonalizable shape operator implies the existence of a path through the origin with noncomparable principal curvatures.

Author Bio

Matthew M. Lukac is a 2014 graduate of the University of Arkansas. He received a B.S. with highest honors in mathematics and physics and plans on attending graduate school to pursue a Ph.D. in mathematics starting Fall 2015. When not doing math, he enjoys mountain biking, long distance cycling, camping, climbing, and cooking. His other interests include metal, video games, and coffee.