Assignment 1; Solutions - PMATH 465/665 Riemannian...

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PMATH 465/665: Riemannian Geometry Assignment 1; Solutions [1] We want to show that every vector field X with compact support is complete. That is, we want to show that the integral curve of X through any p M is defined for all t R . We start with a preliminary lemma. Lemma. Let X Γ( TM ), and let Θ be the flow of X . Suppose there exists an ε > 0 such that, for any point p M , the integral curve Θ ( p ) starting at p is defined for all t ( - ε, ε ). Then X is complete. Proof of lemma. We’ll prove this by contradiction. Suppose there exists a point p M such that the maximal domain D ( p ) of definition of Θ ( p ) is ( a, b ) where either a 6 = -∞ or b 6 = + . The proof is similar in both cases, so let us assume b 6 = + . Choose t 0 ( b - ε, b ), and let q = Θ ( p ) ( t 0 ). By hypothesis, the integral curve Θ ( q ) of X starting at q exists at least for t ( - ε, ε ). Define γ : ( - ε, t 0 + ε ) M by γ ( t ) = ( Θ ( p ) ( t ) , - ε < t < b, Θ ( q ) ( t - t 0 ) , t 0 - ε < t < t 0 + ε. This is well-defined, because in the overlap region ( b - ε, b ) because Θ ( q ) ( t - t 0 ) = Θ t - t 0 ( q ) = Θ t Θ - t 0 Θ t 0 ( p ) = Θ t ( p ) = Θ ( p ) ( t ) by the group law for Θ. Hence γ is an integral curve of X starting at p , and is defined at least until t = t 0 + ε > b , which is a contradiction. We can now easily complete [no pun intended] the argument. Let K = supp( X ). By the ODE theorem, for each p K , there exists some ε p > 0 and some open neighbourhood U p of p such that Θ ( q ) is defined at least on ( - ε, ε ) for all q U p . Since K is compact, the open cover { U p ; p K } of K has a finite subcover { U p 1 , . . . , U p k } . Let ε = min { ε 1 , . . . , ε k } , which is positive. Then Θ ( p ) ( t ) is defined at least for t ( - ε, ε ) for all p K . If p M \ K , then p is a singular point of X , so the integral curve Θ ( p ) is defined for all t R at these points. Hence the hypotheses of the lemma are satisfied, and we conclude that X is complete. Corollary. Let M be a compact manifold. Then any vector field X on M is complete. Proof of corollary. The set supp( X ) is a closed subset of the compact Hausdorff space M , and hence is compact. [2] Let Θ be a flow on an oriented manifold. We want to show that for each t R , the diffeomorphism Θ t is orientation-preserving wherever it is defined. We will begin by deriving an explicit condition for a diffeomorphism f : M N between two oriented manifolds M and N to be orientation-preserving, that we will be able to verify in this case. A diffeomorphism f is orientation-preserving if and only if its pushforward ( f * ) p takes oriented bases of T p M to oriented bases of T f ( p ) N at every p M . If we consider only oriented coordinate charts ( U, ϕ ) and ( V, ψ ) for M and N , then the coordinate frames ∂x 1 p , . . . , ∂x n p of ( U, ϕ ) and ∂y 1 q , . . . , ∂y n q are oriented bases of T p M and T q N , respectively, at every point in their domain.
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