Showing that a map on a tubular neighbourhood is a diffeomorphism

Question

Let $S$ be an embedded submanifold of a smooth manifold $M$. Then there exists a tubular neighbourhood $U$ of $S$ in $M$, that is, $U$ is the diffeomorphic image of $\exp$ restricted to the normal bundle $NS$ of $S$ of a subset $V \subseteq \mathcal{E} \cap NS$, where $\mathcal{E} \subseteq TM$ is the domain of the exponential map, of the form $$V = \{(x,v) \in NS : |v| < \delta(x)\}$$ for some positive continuous function $\delta : S \to \mathbb{R}$.

Let $0 \leq t \leq 1$. Then we can define $\psi_t : U \to U$ by $$\psi_t(\exp(x,v)) := \exp(x,tv).$$

Now the book Introduction to Symplectic Topology by McDuff and Salamon claims that this map is a diffeomorphism for $t > 0$. I do not quite see why. I mean, this map is injective and smooth, but I fail to see surjectivity or the explicit form of an inverse. Moreover, I guess that this map is a local diffeomorphism by considering its derivative. Can anyone help?

Answer 1

After a quick correspondence with the second author, the proof goes as follows: $\psi_t$ is a diffeomorphism onto its image, because $\psi_t$ is injective since $\exp_S$ is. Moreover, an explicit inverse is given by $\exp_S(x, tv) \mapsto \exp_S(x, v) $.