Consider a curved subspace of $\mathbb{R^3}$, in particular an ellipsoid. From an intrinsic geometry point of view, can I tell apart a neighborhood around the pole of the ellipsoid from a neighborhood around the waist?
Is there any topological invariant that would let me tell the neighborhoods apart? If not topological; then any invariant whatsoever?
My attempt at trying to make sense: Being an observer, , with appropriate properties, I would send a circumferential signal $\gamma: \mathbb{R}\to \text{ellipsoid}$ and wait until the signal makes a full revolution and returns to me. All neighborhoods that the curve passes through, I declare to have parameter $t$ to label them. Essentially I need to compactify one spatial dimension, to be able to operationally distinguish the neighborhoods; in order not to wait an infinite amount of time to complete the labeling. The higher the value of the time required for the circle, the closer, this means, I am to the waist.
Is there any better way to understand the global picture from the local? Via the definition of the curvature maybe? The problem with the above is if I need to wait an infinite amount of time for the signal to return to me; i.e., if I can't compactify, then it seems I can't operationally distinguish the neighborhoods?!
This question interests me also because, If I am given a collection of open sets, I need a set of instructions to glue them together and construct the manifold from which they came. This set of instructions I am after.
