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Let $θ$ be a smooth flow on an oriented smooth manifold $M$. I want to show that for every $t\in \mathbb{R}$, $θ_t: M \to M$ is orientation-preserving. This is problem 15-4 from Lee's introduction to smooth manifolds book.

So far I know that since $\theta$ is a smooth flow, $\theta_t$ is a diffeomorphism. How do you go about proving this?

Matt
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    A standard approach is to use the fact that determinant of the relevant map is continuous, $ \theta_t $ is always injective so that the determinant is never zero, and that for $ t = 0 $ you have the identity map, for which the determinant is 1. – user81327 Apr 23 '17 at 17:14
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    Morally speaking, the reason this is true is that $\theta: \mathbb{R} \to \mathrm{Diff}(M)$ is a smooth map (and in fact a smooth group homomorphism), and hence its image is connected. Since $\theta_0= 1_M$ then this curve lies in the orientation preserving component of $\mathrm{Diff}(M)$. – ಠ_ಠ Apr 24 '17 at 00:43
  • @user81327 why is the determinant continuous? – Matt Apr 24 '17 at 03:45

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