Tangent Vectors and Bundles

In $\mathbb{R}^n$ a vector is an arrow, and the derivative of a map is a matrix (2B.8 Calculus in Several Variables). On a curved manifold with no surrounding space, neither makes immediate sense: there is nowhere for the arrow to live. The answer is that each point $p$ of a manifold $M$ has its own vector space of tangent vectors $T_pM$, the possible velocities of curves through $p$, and a smooth map has a differential, a linear map between tangent spaces. In each chart a tangent vector has components, and the components change by the Jacobian matrix when the chart changes, while the vector itself does not.

That last sentence is the key to everything that follows. A tangent vector is a geometric object; its components are a description that depends on the coordinates. The tensors of 8A.7 Tensors and Index Notation, the Riemannian metric among them, are built from tangent vectors and their duals, the covectors, and every formula in the Ricci flow literature is written in components that transform this way.

By the end of this chapter you will be able to:

• define tangent vectors as velocities of curves and as derivations, and work with the coordinate basis $\partial/\partial x^i$;
• compute the transformation law for components under a change of coordinates;
• define and compute the differential of a smooth map, and state the chain rule;
• describe the tangent bundle and vector bundles, trivial and non-trivial;
• define covectors and the differential of a function, and explain why a gradient needs a metric.

3.1The velocity of an aircraft

3.2Tangent vectors

Let $M$ be a smooth $n$-manifold and $p \in M$. Two descriptions of tangent vectors are used, and they agree.

Velocities. For a smooth curve $\gamma : (-\varepsilon, \varepsilon) \to M$ with $\gamma(0) = p$, its velocity $\gamma'(0)$ is defined by what it does to functions: for smooth $f$ defined near $p$,

Two curves have the same velocity when they give the same derivative for every $f$, equivalently when their coordinate representations have the same derivative at $0$ in one (hence every) chart.

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

Every velocity is a derivation, by the product rule. The tangent space $T_pM$ is the vector space of derivations at $p$.

In a chart with coordinates $(x^1, \dots, x^n)$ around $p$, the derivations

are the velocities of the coordinate curves, and they form a basis of $T_pM$ (Lee, chapter 3; the proof uses Taylor's theorem with integral remainder, 2A.11 The Riemann Integral, to show every derivation is determined by its values on the coordinate functions). So $\dim T_pM = n$, and every tangent vector can be written

with the summation convention: an index appearing once up and once down is summed from $1$ to $n$ (8A.7 Tensors and Index Notation). For a curve, $\gamma'(0) = \frac{d\gamma^i}{dt}(0)\frac{\partial}{\partial x^i}$: the components of the velocity are the derivatives of the coordinates.

Example: polar coordinates. On $\mathbb{R}^2\setminus\{0\}$ with $x = r\cos\theta$, $y = r\sin\theta$,

The vector $\partial/\partial\theta$ has Euclidean length $r$, not $1$: coordinate vectors need not be unit vectors, which is why the metric's components $g_{ij}$ appear in every formula later (Exercise 3.8).

3.3The differential

In coordinates $(x^i)$ on $M$ and $(y^a)$ on $N$, with $F$ written as $y^a = F^a(x)$,

the matrix of $dF_p$ is the Jacobian matrix of the coordinate representation (2B.8 Calculus in Several Variables). The chain rule $d(G\circ F)_p = dG_{F(p)}\circ dF_p$ is immediate from the definition. A diffeomorphism has invertible differentials, and conversely the inverse function theorem (2B.9 The Inverse and Implicit Function Theorems) says a map with invertible $dF_p$ is a diffeomorphism near $p$.

3.4Bundles

The tangent bundle is the disjoint union of all tangent spaces,

with the projection $\pi : TM \to M$ sending $T_pM$ to $p$. A chart $(x^i)$ on $U$ gives coordinates $(x^1, \dots, x^n, v^1, \dots, v^n)$ on $\pi^{-1}(U)$, recording the point and the components of the vector; by the transformation law, the transition maps between these charts are smooth, so $TM$ is a smooth manifold of dimension $2n$ (Lee, chapter 3). A vector field is a smooth map $X : M \to TM$ with $X(p) \in T_pM$ for every $p$, written $X = X^i\frac{\partial}{\partial x^i}$ in coordinates (8A.6 Flows and the Lie Derivative).

$TM$ is the basic example of a vector bundle: a smooth family of vector spaces $E_p$ parametrised by $p \in M$, locally a product $U\times\mathbb{R}^k$. A bundle is trivial if it is globally a product $M\times\mathbb{R}^k$. The tangent bundle of $\mathbb{R}^n$, of a torus or of a Lie group (8A.5 Lie Groups and Group Actions) is trivial. The tangent bundle of $S^2$ is not: a trivialisation would give a nowhere-vanishing vector field, which the hairy ball theorem forbids (7A.7 Smooth Topology). The Möbius bundle over the circle, a line bundle whose fibre flips as you go around, is non-trivial for the same reason a Möbius band has one side (7A.1 Topological Spaces and Quotients).

3.5Covectors and the differential of a function

The dual space $T_p^*M = (T_pM)^*$, the linear functions on tangent vectors, is the cotangent space, and its elements are covectors. The basic example is the differential of a function $f$:

In coordinates the differentials of the coordinate functions, $dx^i$, form the dual basis, $dx^i(\partial/\partial x^j) = \delta^i_j$, and

Under a change of coordinates covector components transform the other way from vector components (Exercise 3.4):

This is why indices are placed up for vectors ($v^i$, contravariant) and down for covectors ($\omega_i$, covariant), and why $\omega_iv^i$, summed, is a number that doesn't depend on the coordinates. The cotangent spaces form the cotangent bundle $T^*M$.

A gradient is not a covector but a vector, and turning $df$ into a vector requires an inner product: $\nabla f$ is defined by $g(\nabla f, v) = df(v)$ for all $v$, so in coordinates $(\nabla f)^i = g^{ij}\frac{\partial f}{\partial x^j}$ (8A.7 Tensors and Index Notation, 9A.1 Riemannian Metrics and Model Spaces). In $\mathbb{R}^n$ with the Euclidean inner product and Cartesian coordinates the distinction is invisible, which is why it can be ignored in 2B.8 Calculus in Several Variables. In polar coordinates it is not: $df = f_r\,dr + f_\theta\,d\theta$, but $\nabla f = f_r\,\partial_r + \frac{f_\theta}{r^2}\partial_\theta$.

3.6History

Tangent vectors to surfaces in space go back to Gauss and to the differential geometry of the nineteenth century, where they were arrows in $\mathbb{R}^3$. The idea that a vector at a point of an abstract manifold is an object whose components transform by the Jacobian is Riemann's (1854) and was systematised in the tensor calculus of Gregorio Ricci-Curbastro and Tullio Levi-Civita (1900). The definition of tangent vectors as derivations, independent of any coordinates, and of vector bundles as global objects, was developed in the 1930s–50s by Whitney, Ehresmann, Chevalley and others.

3.7Exercises

Solution

$d\tilde x^j(\partial_i) = \partial_i\tilde x^j$. Substituting, $\omega_idx^i = \omega_i\frac{\partial x^i}{\partial\tilde x^j}d\tilde x^j$ (inverting the relation with the inverse Jacobian). Then $\tilde\omega_j\tilde v^j = \frac{\partial x^i}{\partial\tilde x^j}\omega_i\frac{\partial\tilde x^j}{\partial x^k}v^k = \omega_i\delta^i_kv^k$.

Solution

$dF = \begin{pmatrix}-\ell_1\sin\theta_1 - \ell_2\sin(\theta_1 + \theta_2) & -\ell_2\sin(\theta_1 + \theta_2)\\ \ell_1\cos\theta_1 + \ell_2\cos(\theta_1 + \theta_2) & \ell_2\cos(\theta_1 + \theta_2)\end{pmatrix}$; its determinant simplifies to $\ell_1\ell_2(\sin(\theta_1 + \theta_2)\cos\theta_1 - \cos(\theta_1 + \theta_2)\sin\theta_1) = \ell_1\ell_2\sin\theta_2$. At $\theta_2 = 0$ both columns are multiples of $(-\sin\theta_1, \cos\theta_1)$, so the hand can only move perpendicular to the arm, not along it: a straight arm cannot be lengthened.

Solution

(a) $\frac{\partial}{\partial\tilde x^k} = \frac{\partial x^i}{\partial\tilde x^k}\frac{\partial}{\partial x^i}$, and $g$ is bilinear. (b) $\partial_r = (\cos\theta, \sin\theta)$ and $\partial_\theta = (-r\sin\theta, r\cos\theta)$, so their Euclidean inner products are $1$, $0$ and $r^2$.