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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 $$K(V):=\nabla_{t}v(0).$$

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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).

For the connection map you already have a formula in the definition, just use the facts and get the expression.

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").

The Christoffel symbols of the conformally rescaled metric are given in this Wikipedia article. Using them we get the equations of the conformally related parallel transport.

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See also this answer. – Yuri Vyatkin Mar 29 '14 at 6:28

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