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.
Dr. Phillip Harrington, Department of Mathematical Sciences, University of Arkansas.
Lukac, Matthew M.
"Continuously Diagonalizing the Shape Operator,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 15
, Article 6.
Available at: http://scholar.rose-hulman.edu/rhumj/vol15/iss2/6