Is there a global definition of the tangent bundle?

Question

The tangent bundle of a smooth manifold is usually defined by equipping the disjoint union of the tangent spaces with a smooth structure. Is there a way to define the tangent bundle as a vector bundle first, and then obtain the tangent spaces as the fibres?

Answer 1

I am not sure whether this is what you are after, but I would say that to some extent the description of $TM$ via jets of curves provides such a description. You can define $TM$ as the space $J^1_0(\mathbb R,M)$ of one-jets at $0\in\mathbb R$ of smooth maps $c:\mathbb R\to M$. This has an obvious projection $p:J^1_0(\mathbb R,M)\to M$, defined by $p(j^1_0(c)):=c(0)$.

It is a slightly tricky question whether one knows "in advance" that $p: J^1_0(\mathbb R,M)\to M$ is a vector bundle. In particular, there is the usual problem that the linear structure on tangent spaces is hard to describe in terms of curves. However, for an open subset $U\subset\mathbb R^n$, you immediately see that $J^1_0(\mathbb R,U)\cong U\times \mathbb R^n$ via $j^1_0c\mapsto (c(0),c'(0))$. Using and local charts for $M$, it is easy to prove that $p: J^1_0(\mathbb R,M)\to M$ indeed is a vector bundle.