Continuing on previous posts on affine connection, In this posting we will cover why we called above connection as affine connection.
First review,
affine connection satisfies
In a coordinate basis, affine connection can be defined as
and in a coordinate basis, vector can be defined as
Thus
Under special case such as this connection vanishes we can write
In coordinate transformation such as 
is called affine transformation.
This is due to
where in the process we used
Note that Ax+B is a affine transformation, so that's the reason why we called this connection affine connection.
Furthermore, In Riemannian geometry, such coordinate basis which connection vanishes called Riemann normal coordinate(RNC).
We can do the same thing via last posting
with connection vanishes and solve the equation.