today was my final differential geometry exam and there was a problem that I partially solved, but I have some doubts.
The problem asked to prove that there exists a regular, connected, compact surface $S$ such that the image of the Gaussian curvature is exactly $[0,1]$.
The above implies that all points on the surface are parabolic or elliptical. Since the surface is regular and compact, then it has to be closed (without boundary). Furthermore, there can be no hyperbolic points. Then the requested surface must be something similar to an ellipse. My answer is that the surface must be something like a cylinder to which two halves of a sphere are attached at the ends (drawing attached).
I'm not entirely sure if this surface meets what the exercise asks for. I would like to read your opinions on the problem and if anyone has a way to build such a surface I would appreciate it.