The paper "A micro Lie theory for state estimation in robotics" claims that
the space tangent to $M$ at $X$, which we note $T_X M$. The smoothness of the manifold, i.e., the absence of edges or spikes, implies the existence of a unique tangent space at each point.
I'm wondering why the smoothness of the manifold implies the uniqueness regarding to the tangent space. The confusing part is: what about the two-dimensional surface in Euclidean space? The wiki says that this surface is a two-dimensional manifold. But as far as I'm concerned, the tangent space at each point coincides with each other.
Can someone explain this ?
Update:
In my opinion, any unstructured object is a manifold. In elementary math, we are dealing with structured objects which are exactly what we see everyday in life, e.g. boxes, cups, bands. But as the level of abstractness grows as we're diving into math, there's no way to handle the upcoming problems if we insist on the structured objects. Hence, we define "manifold" that locally resembles Euclidean space. With this characteristic, we're able to apply the rules limited on structured objects, though with some generalization and modification, to everything we see in the math world, without loss of rigorousness.
The tangent spaces are "unique" insofar as the things they contain can be considered different. With regards to a plane (which is a two-dimensional manifold), then the tangent space at every point certainly "seems" the same; and if we are to think of it as the tangent plane at that point of the manifold, it coincides entirely. However, the typical definition of a "tangent vector" at a point is dependent on the point itself, so the different tangent spaces contain "different things" in an abstract sense. Indeed, for a smooth $n$-dimensional manifold, the tangent spaces at any two points will be isomorphic; but they are not isomorphic in any "important" or "useful" way, and it generally does not make sense to suppose they are the same thing.
Intuitively, you might be able to understand this by thinking about the statement that locally, a point's tangent space doesn't care about the "rotation" nor "position" of its tangent plane; thus, the tangent spaces of two points on a plane, considered as a surface, are no more "equal" than the tangent spaces of two points on a $2$-sphere, or two points on any other surface you might care to name.
Immediately after the sentence you mention, the paper goes on to point out that
i.e. the tangent spaces at all points are isomorphic; so it's clear that when the paper says the tangent spaces are "unique" they are talking about something else, and I suspect it is something along the above lines. To summarise, while the use of the word "unique" is perhaps a little confusing, it's definitely correct under the authors' intended interpretation.
In any case, it seems that the main point of that sentence is just to point out that tangent spaces exist everywhere, because the manifold is smooth.