Tangent Spaces and Tangent Bundles

There are several equivalent definitions of the tangent space $T_p M$ at $p \in M$:

Definition (Directional Derivatives). A tangent vector at $p$ is a derivation at $p$: a linear map $v : C^\infty(M) \to \mathbb{R}$ satisfying the Leibniz rule:

$v(fg) = f(p) \cdot v(g) + v(f) \cdot g(p)$

Definition (Equivalence Classes of Curves). A tangent vector is an equivalence class of smooth curves $\gamma : (-\varepsilon, \varepsilon) \to M$ with $\gamma(0) = p$, where $\gamma_1 \sim \gamma_2$ if $(\varphi \circ \gamma_1)'(0) = (\varphi \circ \gamma_2)'(0)$ in some (hence every) chart.

Proposition 2.1. $T_p M$ is a vector space of dimension $n = \dim M$.

The tangent bundle of $M$ is

$TM = \bigsqcup_{p \in M} T_p M = \{(p, v) : p \in M,\ v \in T_p M\}$

It is a smooth manifold of dimension $2n$ with projection $\pi : TM \to M$ given by $\pi(p, v) = p$.

A vector field on $M$ is a smooth section of $TM$: a smooth map $X : M \to TM$ with $\pi \circ X = \mathrm{id}_M$, written $X(p) = X_p \in T_p M$.

Let $f : M \to N$ be a smooth map. The differential (or pushforward) of $f$ at $p$ is the linear map:

$df_p : T_p M \to T_{f(p)} N, \quad df_p(v)(g) = v(g \circ f)$

for $v \in T_p M$ and $g \in C^\infty(N)$. In local coordinates, $df_p$ is represented by the Jacobian matrix $[D(f \circ \varphi^{-1})](\varphi(p))$.

Proposition 2.2 (Chain Rule). $d(g \circ f)_p = dg_{f(p)} \circ df_p$.

In local coordinates $(x^1, \ldots, x^n)$ around $p$, the coordinate vectors $\partial/\partial x^i|_p$ form a basis of $T_p M$. Any tangent vector $v \in T_p M$ can be expressed as:

$v = v^i \frac{\partial}{\partial x^i}\bigg|_p$

For a smooth function $f : M \to \mathbb{R}$, the action of $v$ on $f$ is:

$v(f) = v^i \frac{\partial f}{\partial x^i}(p)$

Example 2.1. On $\mathbb{R}^n$, $T_p \mathbb{R}^n \cong \mathbb{R}^n$ with basis $\partial/\partial x^1, \ldots, \partial/\partial x^n$. A tangent vector $v = (v^1, \ldots, v^n)$ acts on $f \in C^\infty(\mathbb{R}^n)$ by $v(f) = \sum v^i \partial f/\partial x^i(p)$.

Example 2.2. On the sphere $S^2$ at a point $p = (\theta_0, \phi_0)$, the tangent space $T_p S^2$ is spanned by $\partial/\partial\theta$ and $\partial/\partial\phi$ evaluated at $p$.

If $(x^1, \ldots, x^n)$ and $(y^1, \ldots, y^n)$ are two coordinate systems around $p$, the transition formula for tangent vectors is:

$\frac{\partial}{\partial y^j} = \frac{\partial x^i}{\partial y^j} \frac{\partial}{\partial x^i}$

Thus the components of a vector $v = v^i \partial/\partial x^i = \tilde v^j \partial/\partial y^j$ transform as:

$\tilde v^j = v^i \frac{\partial y^j}{\partial x^i}$

This contravariant transformation law characterizes tangent vectors: their components transform using the Jacobian of the coordinate change.

Example 2.3. In polar coordinates $(r, \theta)$ on $\mathbb{R}^2$, the relation between Cartesian and polar basis vectors is:

$\frac{\partial}{\partial r} = \cos\theta\,\frac{\partial}{\partial x} + \sin\theta\,\frac{\partial}{\partial y}, \quad \frac{\partial}{\partial\theta} = -r\sin\theta\,\frac{\partial}{\partial x} + r\cos\theta\,\frac{\partial}{\partial y}$

The dual space $T_p^* M = (T_p M)^*$ is called the cotangent space at $p$. Its elements are covectors (linear functionals on $T_p M$). The basis dual to $\{\partial/\partial x^i\}$ is denoted $\{dx^i|_p\}$, where:

$dx^i\left(\frac{\partial}{\partial x^j}\right) = \delta^i_j$

Proposition 2.3. For a smooth function $f : M \to \mathbb{R}$, the differential $df_p \in T_p^* M$ is given in coordinates by:

$df_p = \frac{\partial f}{\partial x^i}(p)\, dx^i|_p$

The tangent bundle $TM$ is a special case of a vector bundle: a smooth manifold $E$ with a surjective submersion $\pi : E \to M$ such that each fiber $\pi^{-1}(p)$ is a vector space, and local trivializations exist.

Definition. A vector bundle of rank $k$ over $M$ is a smooth manifold $E$ with a smooth map $\pi : E \to M$ such that for every $p \in M$ there exists a neighborhood $U$ and a diffeomorphism $\Phi : \pi^{-1}(U) \to U \times \mathbb{R}^k$ with $\pi = \mathrm{pr}_1 \circ \Phi$ and each fiber $\pi^{-1}(p)$ maps linearly to $\{p\} \times \mathbb{R}^k$.

Example 2.4. The tangent bundle $TM$ is a rank $n$ vector bundle over $M$. The cotangent bundle $T^*M$ is also a rank $n$ vector bundle, dual to $TM$.

Example 2.5. The trivial bundle $M \times \mathbb{R}^k$ is a rank $k$ vector bundle. A manifold $M$ is parallelizable if $TM \cong M \times \mathbb{R}^n$. For example, $S^1$ is parallelizable but $S^2$ is not (by the hairy ball theorem).

Problem 1. Let $M = \mathbb{R}^2$ with coordinates $(x, y)$ and let $f : \mathbb{R}^2 \to \mathbb{R}$ be $f(x, y) = x^2 + y^2$. Compute $df_{(1,0)}$ in coordinates.

Solution. $\partial f/\partial x = 2x$, $\partial f/\partial y = 2y$. At $(1,0)$, $df_{(1,0)} = 2x\, dx + 2y\, dy|_{(1,0)} = 2\, dx$. So $df_{(1,0)}(v) = 2v^1$. $\blacksquare$

Problem 2. Show that $T_p S^2$ is isomorphic to $\{v \in \mathbb{R}^3 : p \cdot v = 0\}$.

Solution. Consider the embedding $S^2 \subseteq \mathbb{R}^3$. A curve $\gamma(t)$ on $S^2$ satisfies $\gamma(t) \cdot \gamma(t) = 1$. Differentiating: $\gamma'(0) \cdot p = 0$, so every tangent vector is orthogonal to $p$. Conversely, any $v \perp p$ is tangent to the great circle in the $p$-$v$-plane. Thus $T_p S^2 \cong p^\perp$. $\blacksquare$

1. Prove that the curve and derivation definitions of $T_p M$ are equivalent.
2. Compute the transition matrix for $T_p \mathbb{R}^2$ between Cartesian and polar coordinates.
3. Show that $d(fg)_p = f(p) dg_p + g(p) df_p$.
4. Prove that $T(S^1)$ is diffeomorphic to $S^1 \times \mathbb{R}$.
5. Show that if $M$ is an $n$-dimensional manifold, then $TM$ is a $2n$-dimensional manifold.