I am new to differential geometry and I learn by myself. It seems that we need something extra called a connection to parallel transport a vector along a curve.
But, suppose we have a vector field $X $, the vector field induces some flow $\phi_t $ on the manifold, which in turn can be viewed as a diffeomorphism of the manifold. Therefore, we can push forward the vector! Can we consider this as a parallel transport of the initial vector?
If not, what is the drawback?