Certain Conditions for a Surface to Be an Oblate Ellipsoid in Three‐Dimensional Euclid Space

In this note, we first demonstrate that, aside from planes, oblate ellipsoids are total umbilical surfaces in the three‐dimensional Euclidean space under a general Euclidean metric, and vice versa. We then generalize the total torsion theorem of spheres to oblate ellipsoids. It is proven that the total torsion of regular closed curves on an oblate ellipsoid in equals zero; conversely, if a surface ensures the total torsion of all regular closed curves in is zero, then it must be a part of an oblate ellipsoid or a plane. Compared with the classical total torsion theorem, our results reveal a deterministic relationship between total umbilical surfaces and metrics.