Reconciling 2 Definitions of Tangent Vectors/Spaces. The Tangent Bundle & The Cotangent Space. (19/09/2025)

The first part of the post is based upon my reading of Arnold’s ODE, the section: Differential Equations on Manifolds; the second part of the post is based upon my reading of Wald’s General Relativity / concepts from class. The motive of this post is to write out my own understanding of these concepts and to reconcile the different definitions that the 2 books give (I believe Arnold approaches it from a more geometric perspective whereas Wald approaches it from a more abstract point of view).

Part 1: Arnold’s ODE

Tangent Space

Let M be a smooth manifold.

Definition of a tangent vector to M at point p: it is an equivalence class of curves passing through p. We say that two curves are equivalence if their images on any chart are equivalent in $\mathbb{R}^n$ (easy to see that this is indeed an equivalence relation).

Recall: the equivalence relation in $\mathbb{R}^n$ is given by: $\gamma_1(t) \tilde \gamma_2(t)$ if the distance between these two curves is $o(|t|)$ as $t \rightarrow 0$ (if we write down a local coordinate, then this equivalence class becomes a vector).

Note: the equivalence relation on M is independent of charts because the transition maps are diffeomorphisms.

Tangent Bundle

We consider $TM = \cup_{p \in M} T_{p}M$ the union of the tangent spaces at each point of M.

TM has a natural smooth manifold structure.

It makes sense for the dimension of TM to be 2n. A point in TM = ‘point of attachment on M’ + ‘direction vector of the tangent space of M at that point’. i.e. to determined a point, we need to know which tangent space it belongs to (the p in $T_p M$ ) – determined by n values since M is dimension n – and then the vector in that tangent space – determined by another n values since $T_p M$ is dimension n.

Figure taken from Arnold’s ODE, last chapter.

We construct a chart on TM from an existing chart on M.

Let $(W, \phi)$ be a chart on M and $(x_1, ..., x_n)$ be the corresponding local coordinates given on $U = \phi(W) \subset \mathbb{R}^n$ . We map each $T_p M$ to the corresponding tangent space of $\mathbb{R}^n$ at $\phi(p)$ by the inverse of the map $\psi : T := \{ \text{equivalence class of curves emanating from} \phi(p) \text{at the instant} t_0 \} \rightarrow \phi(T)$ (this map, as mentioned above, is well-defined and bijective and invariant under the changing of charts).

Note: using $\psi^{-1}$ we can also define a local coordinate on each $T_p M$ given a local coordinate on $T_{\phi(p)} \mathbb{R}^n$ . Let $(\xi_1, ..., \xi_n)$ is the local coordinate on $T_{\phi(p)} \mathbb{R}^n$ . Suppose $\gamma_i : I \rightarrow \mathbb{R}^n$ satisfies $\frac{d \gamma_i}{dt}|_{t_0} = \xi_i$ . Then, $\frac{d \phi^{-1}(\gamma_i)}{dt}|_{t_0}$ gives us the corresponding coordinate on $T_p M$ .

Let $TW = TM|_{p \in W}$ , and $\sigma = (\phi, \psi^{-1})$ . Then $(TW, \sigma)$ , where $\sigma : TW \rightarrow \mathbb{R}^{2n}, \sigma(\textbf{v}) = (x_1, ..., x_n, \xi_1, ..., \xi_n)$ is a chart on TM. We can apply this construction, extending each $(W_i, \phi_i)$ on M to obtain an atlas for TM.

Consistency of charts:

Suppose $(\tilde{TW}, \tilde{\sigma})$ is another chart of M. By construction, $\tilde{\sigma} = (\tilde{phi}, \tilde{\psi}^{-1})$ where $\tilde{\phi}$ is a chart from the atlas of M and $\tilde{\psi}$ is the corresponding map from $T_{\phi(p)}(\mathbb{R}^n)$ to $T_p M$ . By assumption $\tilde{\phi} \circ \phi^{-1}$ is smooth and invertible (this is just a transition map of M).

What about $\tilde{psi}^{-1} \psi$ ?

Let $(x_1, ..., x_n)$ and $(y_1, ..., y_n)$ be the two local coordinates on $\phi(W \cap \tilde{W})$ and $\tilde{\phi}(W \cap \tilde{W})$ , respectively. And:

$y_i = \tilde{\phi} \circ \phi^{-1}_{i}(x_1, ..., x_n)$ .

Let $(\xi_1, ..., \xi_n)$ and $(\eta_1, ..., \eta_n)$ be the two local coordinates on $T_\phi(p) (\mathbb{R}^n)$ for some $p \in W \cap \tilde{W}$ corresponding to $\psi^{-1}$ and $\tilde{\psi}^{-1}$ , respectively. Since these two coordinates arise from the $\phi, \tilde{\phi}$ maps, rewriting the above in coordinates, we have:

$\eta_i = \sum_{k=1}^{n} \frac{\partial \tilde{\phi} \circ \phi^{-1}_{i}}{x_k} \xi_k$

these are smooth functions of the $\xi_k$ ‘s.

Definition: the manifold TM is the tangent bundle of M.

We have 2 natural mappings between M and TM:

$i: M \rightarrow TM$ given by mapping each point $p \in M$ to the zero vector of $T_p M$ .
$p: TM \rightarrow M$ given by the projecting mapping which takes each $T_p M$ and maps it to $p$ .

Definitions:

Fibers: pre-images of points under the projection mapping p. Example: $T_p M$ is a fiber of the bundle TM.
M is the base of TM.

Part 2

Tangent Space

In Wald’s General Relativity, he defines the tangent vector $X_p$ of a smooth manifold at point p to be:

a linear map $X_p : C^{\infty}(M) \rightarrow \mathbb{R}$ satisfying the Leibniz rule, i.e. for any $f, g \in C^{\infty}(M)$ , $X_p(f \cdot g) = X_p(f) \cdot g(p) + f(p) \cdot X_p(g)$ .

So how do we reconcile this definition with the one given by Arnold? How do we see the ‘vector’ hidden behind this definition?

Key: $X_p$ is the map that arises when we differentiate a smooth function along the tangent vector, say $v = (X_1, ..., X_n)$ . Suppose $(x_1, ..., x_n)$ to be the local coordinates at p on M. Then, $X_p = \sum_{k=1}^n X_k \frac{\partial}{\partial x_k}$ .

The equivalency of these two definitions come from the following bijective map:

$(X_1, ..., X_n) \rightarrow X_1 \frac{\partial}{\partial x_1} + ... + X_n \frac{\partial}{\partial x_n}$ .

Recall: this is essentially the one-to-one correspondence between a vector field, $v$ , and its first-order homogeneous linear differential operator, $L_v$ , that Arnold discussed in chapter 1 of his PDE book (and also in chapter 2 of his ODE book).

We are really seeing the tangent space as the space of first-order homogeneous linear differential operators!

In this case, the coordinate tangent vectors corresponding to $(x_1, ..., x_n)$ of the local coordinates on M are: $(\frac{\partial}{\partial x_1}|_p, ..., \frac{\partial}{\partial x_n}|_p)$ and they span the tangent space $T_p M$ .

Cotangent Space

Definition: ${T_p}^{*} M$ is the space of linear functionals $\omega : T_p M \rightarrow \mathbb{R}$ .

This is the dual space of $T_p M$ . It is the space of differential 1-forms.

But how do we see the duality?

Let $df_p \in {T_p}^{*} M$ , we want $df_p(X_p) = X_p(f)$ for any $X_p = (X_1, ..., X_n) \in T_p M$ (or $X_p = \sum_{k=1}^n X_k \frac{\partial}{\partial X_k}$ ).