I don't come from a Differential Geometry background but I have been trying to read a bit about Lie algebras. I am using Humphrey's as my main source but just to get a glimpse of the correspondance between Lie groups and Lie algebras, I am using a particular set of course notes. However, there is something that I cannot really understand about tangent spaces.
Firstly, we will make a major simplification by saying that our manifold $M$ lives inside some $\mathbb{R}^n$. (i.e. $M\subset \mathbb{R}^n$)
For a given point $m\in M$, the tangent space that I am given is:
$T_mM=\{v\in R^n: \exists \textrm{ curve }\gamma:(-\epsilon,\epsilon)\to M \textrm{ with } \gamma(0)=m\textrm{ and } \frac{d}{dt}\gamma(t)|_{t=0}=v\}$
I kind of convinced myself that this definition could be generalised to one involving charts by replacing the last part with $\frac{d}{dt}\phi\gamma(t)|_{t=0}=v$ where $\phi$ is a chart on the manifold containing $m$.
It was left as an exercise to show that $T_mM$ is a vector space. Naively, I thought given $v=\frac{d}{dt}{\gamma_1}|_{t=0}$ and $w=\frac{d}{dt}{\gamma_2}|_{t=0}$, then we have $\gamma=\frac{1}{2}[\gamma_1(2t)+\gamma_2(2t)]$ as a curve that has derivative $v+w$ and $\gamma(0)=m$. But clearly, there is a huge problem that this $\gamma$ may not be a curve in $M$ as $M$ need not be a vector space and in general space manifolds are not vector subspaces of $\mathbb{R}^n$.
I did a bit of searching around and found the definitions to be quite different and defined linear maps called tangent vectors. Now, clearly, linear maps will sidestep this problem about not being a vector space.
How can I modify this definition to circumvent this problem?
(The notes I was using is http://pi.math.cornell.edu/~dmehrle/notes/partiii/liealg_partiii_notes.pdf Page 10)
An equivalent definition of tangent space at a point is to use a chart and its derivative. First we recap on the notion of charts.
Let $p$ be a point on the manifold $M$, and let $\phi:U\to V$ be a chart from $U\subseteq M$ to $V\subseteq\Bbb R^m$, with $U$ open in $M$, $V$ open in $\Bbb R^m$. By definition of a chart, $\phi$ is a homeomorphism from $U$ to $V$, and its inverse $\phi^{-1}$ is smooth as a map $\phi^{-1}:V\to\Bbb R^n$ (remember, $M\subseteq\Bbb R^n$ so we can see the codomain of $\phi^{-1}$ as $\Bbb R^n$ just fine), and the derivative of $\phi^{-1}$ at any point is an injective linear map. Write the derivative of $\phi^{-1}$ at $\phi(p)$ as $(d\phi^{-1})_{\phi(p)}:\Bbb R^m\to\Bbb R^n$.
The following proposition gives you an easy way to show that the tangent space is a vector space.
You may try to prove this yourself. If you cannot, I can provide a reference or a proof sketch.