Elements of a Tangent Bundle

Question

Is an element of a tangent bundle a tangent vector, say an equivalence class of curves, or a vector space where the vectors are equivalence classes of curves. If it's the former (which is my understanding) why bother defining $TM=\sqcup_{p\in M}T_pM$. Note, I know there's a reason but the book I'm reading doesn't really make it clear.

Answer 1

There are a whole lot of different definitions of the tangent bundle. For manifolds that happen to be sitting in euclidean space, a nice definition is that the tangent "plane" at a point $p$ is consists of all vectors $v$ that are the tangent to $M$ at $p$. (Various ways of defining that last notion all give more or less equivalent things.) But to keep things clear, rather than just saying "vectors", we say "based vectors", where a based vector is a pair $(p, v)$ consisting of a basepoint and a vector.

Now it's clear that for the unit sphere, for instance, the vector $(1,0,0)$ is tangent to the sphere at the north pole $n$ and at the south pole $p$. But the based vectors

$$ (n, (1,0,0)) \\ (s, (1,0,0)) $$ are evidently distinct (for their "basepoints" are distinct).

WHY do we do this? Well, it keeps you from talking about the vector $(1,0,0)$ at some point like $(1, 0, 1)$ of the equator; as a vector in 3-space, it's the same as the one that was tangent at the north and south poles, but it's not tangent at $(1,0,1)$.

The idea is that each of these tangent planes is a distinct entity. The (mild) surprise is that all these distinct entities can be gathered together into a structure where "nearby" planes are related in a way that lets you define things like "a continuous field of tangent vectors on the sphere" (where "continuous" is the tricky bit).