# Parallel transport for a conformally equivalent metric

Suppose $M$ is a smooth manifold equipped with a Riemannian metric $g$. Given a curve $c$, let $P_c$ denote parallel transport along $c$. Now suppose you consider a new metric $g'=fg$ where $f$ is a smooth positive function. Let $P_c'$ denote parallel transport along $c$ with respect to $g'$. How are $P_c$ and $P_c$ related?

A similar question is: let $K:TTM \rightarrow TM$ denote the connection map associated to $g$ and $K'$ the one associated to $g'$. How are $K$ and $K'$ related?

In case it's helpful, recall the definition of $K$: given $V\in T_{(x,v)}TM$, let $z(t)=(c(t),v(t))$ be a curve in $TM$ such that $z(0)=(x,v)$ and $\dot{z}(0)=V$. Then set $$K(V):=\nabla_{t}v(0).$$

Both the parallel transport and the connection map are determined by the connection, in your case this is the Levi-Civita connection of metric $g$ whose transformation is known (see e.g. this answer).
With regards to the parallel transport I guess the best way would be to start with the equations $$\dot{V}^{k}(t)= - V^{j}(t)\dot{c}^{i}(t)\Gamma^{k}_{ij}(c(t))$$ that describe the parallel transport (see the details e.g. in J.Lee's "Riemannian manifolds. An Introduction to Curvature").