Telling the neighborhoods of a smooth manifold apart.

Question

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.

Answer 1

The ellipsoid is homeomorphic to the sphere. There's no topological invariant that can tell parts of the sphere apart. The two are even diffeomorphic as smooth manifolds, so that level of structure doesn't get us anything.

If we want to tell points on the ellipsoid apart, we need to go all the way to Riemannian geometry. That lets us define curvature, which is invariant under isometry (of the Riemannian metric) and varies from point to point on the ellipsoid.