Randell 3.3 Defn: A flow on M is a smooth action of the Lie group R^{1} on M, σ: R^{1} × M → M. A flow is also called a dynamical system. σ(t, m) = σ_{t}(m) σ_{0}(m) = m, σ_{t} ∘ σ_{s}(m) = σ_{t+s}(m) = σ_{s} ∘ σ_{t}(m) σ_{−t} = σ_{t}^{−1} σ_{t} : M → M is a diffeomorphism. Ex: σ : R × R^{2} → R^{2}, σ_{t}(x, y) = (x, y) + t(1, 2)

Defn: The orbit of x ∈ M = R(x) = {y ∈ M | ∃t ∈ R such that y = tx} A flow line is the smooth path α_{p} : R → M, α_{p}(t) = σ(t, p). Prop: each q ∈ M lies on a unique flow line.

“differentiating along the flow”: Given a flow σ on M, define s_{σ}: M → T M by s_{σ}(p) = (p, dα_{p}/dt| _{t=0})

Proposition 3.3.3: s_{σ} is a section of T M.

Randell 3.4 The bracket of two vector fields. C^{∞}(M) = {g | g^{smooth} : M → R} Defn: A vector field or section of the tangent bundle T M is a smooth function s: M → T M so that π ∘ s = id [i.e., s(p) = (p, v_{p})]. I.e, s takes p ∈ M to the derivation v_{p} : C^{∞}(M) → R Let f ∈ C^{∞}(M) Define s_{f} : M → R, s_{f}(p) = v_{p}([f]) where s(p) = (p, v_{p}) Note s_{f} is smooth. Thus we can think of a vector field as a function S : C^{∞}(M) → C^{∞}(M), S(f) = s_{f} Lemma 3.4.1: For any vector field s and smooth functions f and g on M, we have

s_{fg}(p) = f(p) · s_{g}(p) + s_{f}(p) · g(p)

Proof: v_{p}(fg) = f(p)v_{p}(g) + v_{p}(f)g(p)

Lemma 3.4.2: Let S : C^{∞}(M) → C^{∞}(M) be linear, and suppose S(fg)(p) = f(p) · S(g)(p) + S(f)(p) · g(p). Then S is a vector field.

Proof: Define S(p) : C^{∞}(M) → R to be the function which sends f ∈ C^{∞}(M) to S(f)(p), i.e, the function S(f) evaluated at p.

Note that the hypothesis implies that S(p) is linear and satisfies the Liebniz rule and hence is a derivation.

Defn: If A, B are vector fields, let AB = A ∘ B

Defn: The Lie Bracket of vector fields A and B is [A, B] = AB − BA : C^{∞}(M) → C^{∞}(M).

Thm: The Lie bracket of vector fields is a vector field.