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I am trying to prove the following equation for $R \in SO(3)$: \begin{equation*} R^{-1}P_{R(\gamma)}(R(v)) = P_{\gamma}(v) \end{equation*} where $\gamma \colon \lbrack 0,1 \rbrack \longrightarrow S^2$ is a curve and $P_{\gamma}$ the parallel transport along $\gamma$.

I have some kind of proof, but I think it can't be correct, because it only uses the fact that $R \colon S^2 \longrightarrow S^2$ is well defined and that $R$ is a diffeo. Here is a sketch:

I used the fact, that the parallel transport is uniquely determined by the unique parallel vector field $X$ with $X(0) = v$ and $\nabla_t X = 0$ on $\lbrack 0,1 \rbrack$. So I chose a map $(x,U)$ around $p$ and wrote down the differential equations in the basis belonging to the map $(x \circ R^{-1}, R(U))$, which is just given by $R(X_i)$, when $X_i$ are the basis vectors for $T_p S^2$ corresponding to $x$ (is that correct?). Then I could use linearity of $R$ to show that if I multiply with $R^{-1}$ the differential equations for the parallel transport $P_{\gamma}$ are fullfilled.

Something must be wrong, because I dont use $\det(R) = 1$ at all. Any hints, where this is needed, would be very helpfull..

Thanks a lot.

TerranMathematician
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1 Answers1

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The differential equation for parallel transport involves the Riemannian metric. So your proof will have to use the fact that $R$ preserves the metric (i.e. that if $x \in S^2$ and $v,w$ are tangent vectors at $x$, then $g_{R(x)}(R_*v, R_*w) = g_x(v,w)$). On the other hand, this is all that will be needed (together with the fact that $R$ is smooth, so that it respects the various derivatives and so on that are involved).

The fact that det $R = 1$ isn't important; a reflection about a plan through the origin will also preserve parallel transport (since it preserves the metric). The key thing is that $R$ does in fact preserve the metric on $S^2$, which in turn follows from the fact that $R$ is the restriction to $S^2$ of an isometry of $\mathbb R^3$. (You might want to first check that $R$ preserves the Riemannian metric on $\mathbb R^3$, which will follow essentially by definition. Then check that if $R$ is an isometry of some ambient manifold that preserves a submanifold $M$, then it preserves the induced metric on $M$.)

Matt E
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