T M = ∪_{p∈M}T_{p}(M) = {(p, v) | p ∈ M, v ∈ T_{p}M}, let π: T M → M be defined by π(p, v) = p. Let (ϕ, U) be a chart for M. If q ∈ U, let \{(\frac{∂}{∂x_{1}} )_{q}, ..., (\frac{∂}{∂x_{m}})_{q}\} be a basis (w.r.t (ϕ, U)) for T_{q}(M) = T_{q} t_{ϕ}: π^{−1}(U) → ϕ(U) × R^{m} ⊂ R^{2m}, t_{ϕ}(q, v) = (ϕ(q), a_{1}, ..., a_{m}) where v = Σ_{i=1}^{m}a_{i}(\frac{∂}{∂x_{i}})_{q} Let A be a maximal atlas for M. Basis for topology on T M : {W | ∃(ϕ, U) ∈ A s.t. W ⊂ π^{−1}(U) and t_{ϕ}(W) open in R^{2m}} Claim: T M is a 2m−manifold and C = {(t_{ϕ}, π^{−1}(U)) | (ϕ, U) ∈ A} is a pre-atlas for T M. π: T M → M, π(p, v) = p is smooth df : T M → T N defined by df(p, v) = (f(p), d_{p}f(v)) is smooth if f : M → N is smooth. Proof: See Hitchin 4.1 (in Chapter 1 of http://www2.maths.ox.ac.uk/ hitchin/hitchinnotes/hitchinnotes.html 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})]. Defn: s is never zero if s(p) ≠ (p, 0) for all p ∈ M. Prop: Let G be a Lie group. Then G admits a never-zero vector field. Note: S^{n} admits a never-zero vector field iff n odd. Let p_{2}(s(p)) = p_{2}(p, v_{p}) = v_{p} Defn: The vector fields s_{1}, ..., s_{k} are linearly independent iff for all p ∈ M, p_{2}(s_{1}(p)), ..., p_{2}(s_{k}(p)) are linearly independent. Prop: M is parallelizable (or equivalently the “tangent bundle π: T M → M is trivial”) iff T M admits m linearly independent vector fields. 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. A flow line is the smooth path α_{p} : R → M, α_{p}(t) = σ(t, p).