# Parallel transport as a mapping between vector spaces

I have a general question about the nature of parallel transport. As far as I understand it, given a tensor at point $p$ and a smooth curve going through $p$, the parallel transport equations determine a tensor at any other point $q$ along the curve.

My question is, does the parallel transport equation simply associate, for each point $q$ along the curve, a tensor in the vector space at point $p$ with the point $q$, or does it actually map the tangent space at $p$ into the tangent space at $q$.

Perhaps more fundamentally, given a chart and two points $p$ and $q$ in said chart, for any tensor in $V_p$ is there a tensor in $V_q$ with the same components, and if so, are the tangent spaces only fundamentally different because they are based at different points on the manifold?

• If no answers appear in a few days, perhaps you could consider posting this at Mathematics. Commented Dec 8, 2017 at 10:58
• Parallel transport doesn't typically preserve the components. Commented Dec 8, 2017 at 13:33
• On a semi-Riemannian manifold, the natural coordinate vector fields are parallel and so are their restirctions to a curve. Hence parallel translation from $p$ to $q$ along any curve is a canonical isomorphism $v_{p} \rightarrow v_{q}$. Commented Aug 19, 2019 at 3:33
• You might find the wiki page on holonomies useful. Commented Dec 24, 2021 at 15:24

Perhaps more fundamentally, given a chart and two points $p$ and $q$ in said chart, for any tensor in $V_p$ is there a tensor in $V_q$ with the same components, and if so, are the tangent spaces only fundamentally different because they are based at different points on the manifold?
This I believe answers your question, but for completeness parallel transport is a specialized general linear map between these tangent spaces insofar that its differential version is a covariant derivative along the tangent to the curve defined by a connection. In its most abstract form, a connection is simply a linear map that (1) assigns to a smooth section $\Gamma(T\,M)$ of the tangent bundle $T\,M$ (i.e. a vector field on the manifold $M$) another smooth section $\Gamma(T\,M \otimes T^* M)$ of the tangent bundle tensored with the cotangent bundle $T^* M$ and (2) fulfills the Leibniz rule when we multiply our smooth section by any smooth function. That $T\,M \otimes T^* M$ looks fearsome, but all we are saying is that we begin with a vector field $X$ and our function returns a vector-field-valued linear function of $\nabla_Y\,X$ of a second vector field $Y$ - the second argument is simply where we put the "direction" we want to differentiate along. When we're calculating parallel transports along a system of curves, the vector field $Y$ is the system of tangents to these curves (or, contrariwise, the curves are the flow of the vector field $Y$) and we put $Y$ as the argument into our covariant derivative $\nabla_Y X$ to work out the directional derivative of $X$ along $Y$. The Leibnitz rule condition is simply that $\nabla_Y (f\,X) = \nabla_Y X + \nabla_Y f \otimes X$ for any smooth function $f:M\to\mathbb{R}$ defined on the manifold.