Hi i am reading An introduction to manifolds by Loring and have some doubts in proposition 8.16. The statement that i have doubt in is:
By proposition 8.15, the simplest such $\alpha$ is $\alpha(t)=(a^1 t,...,a^n t)$
My questions are as following:
- How did we get $\alpha(t)=(a^1 t,...,a^n t)$ using proposition 8.15? What are $\dot {c}^i(t)$ and $\left.\frac{\partial}{\partial x^i}\right|_{c(t)}$ in this case?
- Why is it the simplest such $\alpha$? Is it because it is linear in $t$ that is we could have chosen more complicated dependence on $t$ but we have chosen the simplest in this sense.?
- Will the coefficients $a^1,...,a^n$ vary as $t$ varies in $\alpha(t)$ or are those coefficients constant?
For reference i am attaching the screenshots of both the propositions where i have highlighted the part in which i have doubt in.
In the proof of 8.16, in the part right before where you highlighted in pink, the goal is to find a curve $\alpha$ in $\Bbb{R}^n$ with $\alpha'(0) = (a_1,\dots,a_n)$.
Proposition 8.15 tells you that for any curve $\alpha(t) = (\alpha_1(t), \dots, \alpha_n(t))$, the velocity is $\alpha'(t) = (\dot{\alpha}_1(t), \dots,\dot{\alpha}_n(t))$. So you want to find such an $\alpha$ so that $\dot{\alpha}_1(0) = a_1$, $\dot{\alpha}_2(0)=a_2$, $\dots$, $\dot{\alpha}_n(0) = a_n$. I think what the author is saying is that the simplest way to do so is to use a curve for which $\dot{\alpha}_i(t) = a_i$ FOR ALL $t$ (not just $t=0$). In other words, the "simplest" way is to just have $\alpha'(t)$ be constant, and not depend on $t$ at all! This is where you get the linear expressions $\alpha_i(t) = a_i \, t$.
Just to clarify: Proposition 8.15 doesn't say this is the only choice of $\alpha$ that will work. The author is just saying this is what he/she considers the easiest choice which happens to work.