Vector Fields and Flows

Let $X$ be a smooth vector field on $M$. An integral curve of $X$ through $p$ is a smooth curve $\gamma : I \to M$ such that $\gamma(0) = p$ and $\gamma'(t) = X_{\gamma(t)}$ for all $t \in I$.

Theorem 3.1 (Existence and Uniqueness). For every $p \in M$, there exists a unique maximal integral curve $\gamma_p : I_p \to M$ of $X$ through $p$, defined on a maximal open interval $I_p \subseteq \mathbb{R}$ containing $0$.

For vector fields $X, Y$ on $M$, the Lie bracket $[X, Y]$ is the vector field defined by:

$[X, Y](f) = X(Y(f)) - Y(X(f))$

for $f \in C^\infty(M)$.

Proposition 3.2 (Properties of the Lie Bracket).

1. Bilinearity: $[aX + bY, Z] = a[X, Z] + b[Y, Z]$.
2. Anti-symmetry: $[X, Y] = -[Y, X]$.
3. Jacobi identity: $[X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0$.

The space of all vector fields $\mathfrak{X}(M)$ with the Lie bracket forms a Lie algebra.

The Lie derivative of a vector field $Y$ along $X$ is $\mathcal{L}_X Y = [X, Y]$. For a function $f$, $\mathcal{L}_X f = X(f)$.

The Lie derivative measures the rate of change of a geometric object along the flow of $X$.

Theorem 3.3 (Flow of the Lie Bracket). If $\Phi_t$ and $\Psi_s$ are the flows of $X$ and $Y$ respectively, then $\frac{d}{dt}\big|_{t=0} (\Psi_{-t})_*(\Phi_t)_* Y = [X, Y]$.

The flow of a vector field $X$ is a smooth map $\Phi : \mathcal{D} \to M$, where $\mathcal{D} \subseteq \mathbb{R} \times M$ is an open domain, defined by $\Phi(t, p) = \gamma_p(t)$, the integral curve of $X$ through $p$.

Proposition 3.4 (Flow Properties). For each $p \in M$, there exists $\varepsilon > 0$ and a neighborhood $U$ of $p$ such that:

1. $\Phi(0, p) = p$.
2. $\Phi(t, \Phi(s, p)) = \Phi(t+s, p)$ whenever both sides are defined.
3. For each $t$, the map $\Phi_t : U \to M$ defined by $\Phi_t(p) = \Phi(t, p)$ is a diffeomorphism onto its image, with inverse $\Phi_{-t}$.

Definition. A vector field $X$ is complete if its flow is defined for all $t \in \mathbb{R}$ (i.e., $\mathcal{D} = \mathbb{R} \times M$). This happens if the maximal interval $I_p$ is all of $\mathbb{R}$ for every $p \in M$.

Theorem 3.5 (Compactness Implies Completeness). If $M$ is compact, then every smooth vector field on $M$ is complete.

Example 3.1. On $M = \mathbb{R}$, the vector field $X = \partial/\partial x$ is complete with flow $\Phi(t, p) = p + t$. The vector field $X = x^2 \partial/\partial x$ is not complete: the integral curve through $p > 0$ satisfies $\dot\gamma = \gamma^2$, giving $\gamma(t) = 1/(1/p - t)$, which blows up at $t = 1/p$.

A one-parameter group of diffeomorphisms is a smooth map $\Phi : \mathbb{R} \times M \to M$ such that $\Phi_t \circ \Phi_s = \Phi_{t+s}$ and $\Phi_0 = \mathrm{id}_M$.

Proposition 3.6. There is a bijection between complete vector fields on $M$ and one-parameter groups of diffeomorphisms of $M$. Given a complete vector field $X$, its flow $\Phi_t$ is a one-parameter group. Conversely, given a one-parameter group $\Phi_t$, define $X_p = \frac{d}{dt}\big|_{t=0} \Phi_t(p)$.

Example 3.2. On $\mathbb{R}^2$, the vector field $X = -y \partial/\partial x + x \partial/\partial y$ generates rotation: $\Phi_t(x, y) = (x\cos t - y\sin t, x\sin t + y\cos t)$. This is a one-parameter group of rotations.

Proposition 3.7. Two vector fields $X, Y$ have commuting flows if and only if $[X, Y] = 0$.

More precisely, $[X, Y] = 0$ if and only if for all sufficiently small $s, t$: $\Phi_t^X \circ \Phi_s^Y = \Phi_s^Y \circ \Phi_t^X$, where $\Phi^X$ and $\Phi^Y$ are the flows of $X$ and $Y$ respectively.

Example 3.3. On $\mathbb{R}^3$, the vector fields $X = \partial/\partial x$ and $Y = \partial/\partial y$ commute: $[X, Y] = 0$. Their flows are translations in the $x$ and $y$ directions respectively, and these clearly commute.

Example 3.4. On $S^2$, the vector fields generating rotations about the $x$-axis and $y$-axis do not commute: $[X, Y] = Z$, where $Z$ generates rotation about the $z$-axis. This reflects the non-commutativity of the Lie algebra $\mathfrak{so}(3)$.

In local coordinates $(x^1, \ldots, x^n)$, a vector field $X$ can be written as:

$X = X^i(x) \frac{\partial}{\partial x^i}$

The integral curve equation $\dot\gamma(t) = X_{\gamma(t)}$ becomes the system of ODEs:

$\dot\gamma^i(t) = X^i(\gamma(t)), \quad i = 1, \ldots, n$

The Lie bracket in coordinates is:

$[X, Y]^i = X^j \frac{\partial Y^i}{\partial x^j} - Y^j \frac{\partial X^i}{\partial x^j}$

Problem 1. Let $X = x \partial/\partial x$ on $\mathbb{R}$. Find the flow and determine whether $X$ is complete.

Solution. The ODE is $\dot\gamma = \gamma$, giving $\gamma(t) = pe^t$. So $\Phi(t, p) = pe^t$. This is defined for all $t \in \mathbb{R}$, so $X$ is complete. $\blacksquare$

Problem 2. Compute $[X, Y]$ for $X = y \partial/\partial x$ and $Y = x \partial/\partial y$ on $\mathbb{R}^2$. What do their flows look like?

Solution. Using the coordinate formula: $X^1 = y$, $X^2 = 0$, $Y^1 = 0$, $Y^2 = x$.

$[X, Y]^1 = y\frac{\partial(0)}{\partial x} - 0\frac{\partial(y)}{\partial x} + 0\frac{\partial(0)}{\partial y} - x\frac{\partial(y)}{\partial y} = -x$

$[X, Y]^2 = y\frac{\partial(x)}{\partial x} - 0\frac{\partial(0)}{\partial x} + 0\frac{\partial(x)}{\partial y} - x\frac{\partial(0)}{\partial y} = y$

So $[X, Y] = -x \partial/\partial x + y \partial/\partial y$. The flows are: $\Phi_t^X(x,y) = (x+yt, y)$ (shear), $\Phi_s^Y(x,y) = (x, y+xs)$ (shear). These do not commute. $\blacksquare$

1. Find the flow of $X = x^2 \partial/\partial x + y \partial/\partial y$ on $\mathbb{R}^2$.
2. Prove that $[X, fY] = f[X, Y] + (Xf)Y$ for $f \in C^\infty(M)$.
3. Show that the vector field $X = \partial/\partial\theta$ on $S^1$ is complete and find its flow.
4. Compute the Lie bracket of $X = \partial/\partial x$ and $Y = x \partial/\partial y$ on $\mathbb{R}^2$.
5. Prove that if $[X, Y] = 0$ then $\Phi_t^X \circ \Phi_s^Y = \Phi_s^Y \circ \Phi_t^X$ for all $s, t$.