Path Independence of parallel transport

For a vector field to be covariantly constant at a point in a curved space, it is mandatory that theRiemann-Tensor is also zero at that point. Thus, in the presence of curvature, we cannot define a vector corresponding to the field at the point by looking for a covariantly constant vector. Instead, we consider a curve and then define a vector field to be covariantly constant along the curve or parallel transported along the curve if $V^{\lambda }{\nabla }_{\lambda }{W}^{\mu }=0$ The way vectors change under parallel transport around arbitrary closed loops is one way to characterise the curvature of a space -Holonomy