I am reading the note and get confused with some computation in there.
Let $M$ be a Riemannian manifold. Let $f: [a,b]\times (-\epsilon,\epsilon)\rightarrow M$ is a smooth map.
Denote $f_{s} := \dfrac{\partial f}{\partial s}$ and $f_{t} := \dfrac{\partial f}{\partial t}$. So these are vector files along the image of $f$.
Then he claimed that $\nabla_{f_{s}}f_{t} = \nabla_{f_{t}}f_{s}$.
At this point I am confused with what definitions of $\nabla_{f_{s}}f_{t}$ and $\nabla_{f_{t}}f_{s}$ are.
In the proof he wrote that $$\nabla_{f_{s}}f_{t} - \nabla_{f_{t}}f_{s} = [f_s,f_t].\quad (\star)$$
Since we assume the torsion is free, I guess that ($\star$) comes from this.
Question: the torsion tensor is defined over smooth vector fields on $M$, how to define the torsion over $f_s,f_t$, which are only vector fields along the image of $f$.
He then wrote that $$[f_s,f_t] = df[\dfrac{\partial}{\partial s},\dfrac{\partial}{\partial t}].$$
Question: why do we have this formula? How to define the Lie bracket of two vector fields along the image of $f$ only?
