Vector Fields and Flows

3.1 Integral Curves

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 $0$.

3.2 The Lie Bracket

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.

3.3 The Lie Derivative

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 $\frac{d}{dt}\big|_{t=0} (\Psi_{-t})_*(\Phi_t)_* Y = [X, Y]$.