Fiber-preserving diffeomorphism

Question

What does it mean in the definition 7.1(ii) that the diffeomorphism $$\phi_U$$ is fiber-preserving? In what sense/how technically it preserves fibers? They have never defined this in the book before.

Answer 1

The textbook is referring the property of $\phi_U$ restricting to a linear isomorphism between $E_p$ and $p \times \mathbb{R}^r$ as fiber-preserving. The reason behind this terminology is that if we restrict $M$ to $p$, then the following diagram commutes where $pr_1$ is the projection onto the first component of $p \times \mathbb{R}^r$. This is essentially saying that $E$ resembles the base space $M$ when zooming in close enough on $p$.