Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|cite|improve this question

1 Answer 1

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.

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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.