I was doing Lee's smooth manifold book exercise in Problem 15.4 needs to show
Let $M$ be a oriented smooth manifold ,and $\theta$ the flow generated by some smooth vector field.
Prove that $$\theta_t :M_t \to M_{-t}$$ is orientation preserving diffeomorphism.
First it's diffeomorphism by fundamental theorem of flow.To prove that is orientation preserving seems rather complicated,the rough idea is simple we need to prove that Jacobian under positive oriented chart has positive determinant.Formally if all of them lies in the single chart for all time $t\in \Bbb{R}$ and all point $p\in M$ then the Jacobian is $$\frac{\partial \theta^j}{\partial x^i}(t,p)$$ is smooth w.r.t to time $t$ which is defined for all $t$. which connect $0$ and $t$.hence they have same sign of determinate.
The question become rather complicated,the reason is when $t$ varies,the point may varies the chart may vary that Jacobian is not defined in a consistent way.So I have no idea how to deal with this problem?
Best Answer
There is a "coordinates free" way of the above argument.
Let $\omega$ be a non-vanishing $n$-form on $M$ which defines the orientation. Then a diffeomorphism $\Phi : M \to M$ is orientation preserving if and only if $\Phi^*\omega = g\omega$ for some positive function $g$ (since $\Phi$ is a diffeomorphism, $g$ is always non-zero).
In our situation, one may write $$ \theta_t^* \omega = g_t \omega$$ for a family of smooth functions $g_t : M \to \mathbb R$. Since $\theta_0 = \operatorname{Id}$, $g_0 = 1$ is the constant positive function. Thus by continuity, $g_t$ is positive for all $t$.