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We’ll have shown that an integral curve with γ(0) = p exists for all positive time. A simple criterion for excluding ii. and iii. is the following. 56 Chapter 2. 9. The scenarios ii. and iii. can’t happen if there exists a proper C 1 -function, ϕ : U → R with Lv ϕ = 0. Proof. Lv ϕ = 0 implies that ϕ is constant on γ(t), but if ϕ(p) = c this implies that the curve, γ(t), lies on the compact subset, ϕ −1 (c), of U ; hence it can’t “run off to infinity” as in scenario ii. or “run off the boundary” as in scenario iii.

Let Ui , i = 1, 2, be open subsets of Rni , vi a vector field on Ui and f : U1 → U2 a C ∞ -map. If v1 and v2 are f -related, every integral curve γ : I → U1 of v1 gets mapped by f onto an integral curve, f ◦ γ : I → U 2 , of v2 . 14. Suppose v1 and v2 are complete. Let (fi )t : Ui → Ui , −∞ < t < ∞, be the one-parameter group of diffeomorphisms generated by vi . Then f ◦ (f1 )t = (f2 )t ◦ f . Hints: 1. Theorem 4 follows from the chain rule: If p = γ(t) and q = f (p) dfp d γ(t) dt = d f (γ(t)) .

A one-form on U is a function, ω, which assigns to each point, p, of U a vector, ω p , in (Tp Rn )∗ . Some examples: 1. Let f : U → R be a C 1 function. 12) dfp : Tp Rn → Tc R and by making the identification, Tc R = {c, R} = R 54 Chapter 2. , as an element of (Tp Rn )∗ . 13) defines a one-form on U which we’ll denote by df . 2. Given a one-form ω and a function, ϕ : U → R the product of ϕ with ω is the one-form, p ∈ U → ϕ(p)ω p . 3. Given two one-forms ω1 and ω2 their sum, ω1 + ω2 is the one-form, p ∈ U → ω1 (p) + ω2 (p).

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Multilinear algebra and differential forms for beginners by Guillemin V.


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