A couple of questions regarding the Vector Bundle Chart Lemma

Question

I have a couple of questions regarding the proof of the Vector Bundle Chart Lemma in Lee's Introduction to Smooth Manifolds:

• Why is $\mathbb{H}^n\times\mathbb{R}^k\cong\mathbb{R}^{n+k}$, and what is the meaning of $\cong$ here?

• Where is condition (iii) used? As far as I can see the proof only requires $\Phi_\alpha\circ\Phi_\beta^{-1}$ to be a diffeomorphism.

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Answer 1

• $\cong$ means diffeomorphic, and you have a typo; it should be $\Bbb{H}^{n+k}$. Just take the example $n=k=1$. $\Bbb{H}^1\times\Bbb{R}^1=[0,\infty)\times\Bbb{R}$. Compare with $\Bbb{H}^{1+1}=\Bbb{H}^2=\{(x,y)\in\Bbb{R}^2\,:\, y\geq 0\}$. Can you see the obvious diffeomorphism (ok first what’s the obvious bijection? next, why is it a diffeomorphism)? This generalizes almost instantly. Extra FYI: ‘more’ generally, a closed halfspace in a normed vector space $V$ can be specified by a choice of a non-zero $\lambda\in V^*$ by simply considering $H=\lambda^{-1}([0,\infty))$.
• Condition (iii) is used to provide $E$ with a well-defined vector space structure on each fiber. See the second last paragraph. If the transition $\Phi_{\alpha}\circ\Phi_{\beta}^{-1}$ was simply a diffeomorphism which preserves the fibers, then using one $\Phi_{\alpha}$, you’d get one vector space structure, but with another $\Phi_{\beta}$, you’d get another, so there might be incompatibility in the linear structure on each fiber.