What is a tangent bundle? (Aubin)

Question

Here's what I read in A Course in Differential Geometry by Thierry Aubin.

2.5. Definition. The tangent bundle $T(M)$ is $\bigcup_{P\in M} T_P(M).$

And then

2.6. Definition. Let $\Phi$ be a differentiable map of $M_n$ into $W_p$ (two differentiable manifolds). Let $P\in M_n,$ and set $Q=\Phi(P).$ The map $\Phi$ induces a linear map $(\Phi_*)_P$ of the tangent bundle $T_P(M)$ into $T_Q(W)$ defined by $$[(\Phi_*)_PX](f)=X(f\circ\Phi);$$

here $X\in T_P(M),\;(\Phi_*)_PX\in T_Q(W)$ and $f$ is a differentiable function in a neighbourhood $\theta$ of $Q.$ We call $(\Phi_*)_P$ the linear tangent mapping of $\Phi$ at $P.$

I don't understand why the author calls $T_P(M)$ a tangent bundle in the second definition. Is it a mistake? From the first definition, a tangent bundle is the union of all tangent spaces over all points of the manifold. And $T_P(M)$ is just one tangent space, at a particular point $P$.

And an additional question: Should I be worried whether the union in the first definition is disjoint or not? After a moment's thought, I believe it might turn out not to be according to the previous definitions.

Answer 1

Definition 2.6 has a typo; $(\Phi_\ast)_P$ is a map of tangent spaces, not tangent bundles (although all of the $(\Phi_\ast)_P$ combine to form a bundle map between tangent bundles).

The tangent bundle is the disjoint union of the tangent spaces: $$TM = \coprod_{P \in M} T_P M.$$ It has the topology of a smooth manifold in the following manner. Let $(U_\alpha, \phi_\alpha)$ be an atlas for $M$, and let $\pi: TM \longrightarrow M$ be the natural projection, i.e. if $(P, v) \in T_P M \subset TM$, then $\pi(P, v) = P$. Then we get an atlas $(\pi^{-1}(U_\alpha), \tilde{\phi}_\alpha)$ for $TM$, where $$\tilde{\phi}_\alpha(P, v) = (\phi_\alpha(P), v).$$

Answer 2

It looks to me like Aubin isn't assuming you know the general definition of a bundle. So strictly speaking he can't really talk about a "linear map of bundles", only a linear map of each individual tangent space, and he's trying to finesse that point. I'd say you're right that it's poorly phrased, though. (The reason you shouldn't worry too much about it is that a linear map of bundles is pretty much just a linear map on each tangent space separately, with some added continuity/differentiability conditions that are more or less automatic in this case.)

For your additional question, you should definitely take the union in the first definition to be disjoint, or bad things will happen. If it makes you feel better, you can think of $TM$ as consisting of ordered pairs $(p,v)$ where $v \in T_p M$.

Answer 3

I will begin by attempting to address the question in the subject and definition 2.5.

The tangent bundle of a differentiable manifold $M$ is way to organize all of the (point-wise) tangent spaces of $M$ into a formal geometric object (in fact, the tangent bundle $T(M)$ turns out to be a differentiable manifold in its own right). When first learning the material, I would not be too worried about whether you should have a disjoint union or a union in the formal definition; I would be concerned about what the definition is trying to tell you. One way to think of the definition of the $T(M)$ is that it is family of vector spaces parametrized by the manifold $M$. At each point $P \in M$, you get an $n$-dimensional vector space associated to $P$. In this case, the vector space associated to the point $P$ is the tangent space $T_{P}M$. In formal ``bundle''terminology, the tangent space $T_{P}M$ associated to a point $P \in M$ is called the fiber over $P$.

In definition 2.6, the author is trying to tell you that a differentiable map $\Phi : M \to W$ induces a mapping $\Phi_{*} : T(M) \to T(W)$ that maps the fiber over $P$ (i.e. $T_{P}M$) to the fiber over the image of $P$ (i.e. $T_{Q}W$, where $\Phi(P) = Q$). It does appear that the use of the word ``tangent bundle'' in definition 2.6 is not quite correct (or at least, could be confusing). Maybe something along the following lines would work better:

``. . . At each $P \in M$, $\Phi$ induces a linear mapping $(\Phi_{*})_{P}: T_{P}M \to T_{Q}W$ defined by, . . .''

It should be noted that when all of the fiber wise maps are pieced together, however, one does indeed obtain a linear map $\Phi_{*} : T(M) \to T(W)$.

There is a more general definition of a vector bundle over a manifold $M$, of which the tangent bundle is the prototypical example. (I find Spivak's Comprehensive Introduction to Differential Geometry (I forget which volume), or John Lee's Introduction to Smooth Manifolds to be quite helpful on these matters.)