First of all! I mean a cylinder with a circle at the top and a circle at the bottom. Not the volume but its surface.
Intuition tells me YES. But...
How can charts be defined for the points on the edge? What I thought first is that since a sphere is a differentiable manifold and I could find a bijection projecting the cylinder on a sphere placed in its center (of symmetry) I could extend the properties of the sphere to the cylinder.
Now some doubts arise since the metric that I would define on the cylinder would be spherical, while I would expect to get standard euclidean geometry on the flat surfaces. Anyway, no metric is yet defined on differentiable manifolds and I can imagine to define the metric that best fits after. So
- Wthout a metric, does the differential manifold structure allows me to distinguish between a cylinder and a sphere?
- With a metric can I be sure whether my manifold is a cylinder or a sphere or anything else?
- Where can I get a handy summary on the structure information given by topology, manifold, and metric when I only know some of the three? thank you in advance