Ovaloids of 3 and Their Umbilics: A Differential Equation Approach

The following extension of the Carathéodory conjecture on umbilics is proposed and solved—in very special cases: Let N: → 2 be the Gauss map of a C3 ovaloid 3. Let be the set of umbilics q such that N(q)=q. If A is a connected component of , then, each connected component of \A contains at least an umbilic point of .