Let S⊂R3 be a connected smooth surface. Suppose that every point of S is an umbilic point. Prove that S is a subset of either a plane or a sphere in R3
I'm ok with the statement by itself, but i was wondering: can this be a if and only if? Because i just recieved the results of an exam: i had to determine if a surface S was a subset of a sphere or a plane. I found out not all points were umbilical and i stated that the statement above wasn't enough and the teacher's comment was "conclusions?".
Am i missing something? I apologize in advance for any typos i might have done or bad font and thank you