# Is parallel displacement/transport a smooth map?

Let $$G$$ be a Lie group, $$M$$ be a smooth manifold and $$\pi:P\rightarrow M$$ be a principal $$G$$-bundle. Given a connection $$1$$-form $$A:P\rightarrow \Lambda^1_{\mathfrak{g}}T^*P$$ on the principal bundle, fixing a point $$x\in M$$ we have the notion of holonomy map $$\Omega(M,x)\rightarrow \text{Aut}(\text{fiber of P at x})$$

Recall : Given a loop (path) $$\gamma$$ in $$M$$, based at $$x$$, and a point $$u\in \pi^{-1}(x)$$, the connection gives a path $$\gamma^*_u$$ in $$P$$ starting at $$u$$.Varying $$u$$ over $$\pi^{-1}(x)$$, we get paths $$\{\gamma^*_u:[0,1]\rightarrow P|u\in \pi^{-1}(x)\}$$. This gives a map $$\Phi_x:\pi^{-1}(x)\rightarrow \pi^{-1}(x)$$ defined as $$u\mapsto \gamma^*_u(1)$$. I was wondering if $$\Phi_x$$ is a smooth map (recall that $$\pi^{-1}(x)$$ is an embedded submanifold of $$P$$).

Is the parallel transport/displacement map $$\Phi_x:\pi^{-1}(x)\rightarrow \pi^{-1}(x)$$ a smooth map for each $$x\in M$$? I am sure this is true but could not prove it now. Any help is appreciated.

• Of course, you do not even need to assume that the bundle is principal. This should be in volume 1 of Kobayashi-Nomizu. The proof boils down to applying a theorem from ODEs on smoothness of the solution of the Cauchy initial value problem for 1st order ODEs. – Moishe Kohan Sep 23 '19 at 16:13
• @MoisheKohan I have seen in Kobayashi and Nomizu, I do not see mention of smoothness of the map $\tau:\pi^{-1}(x)\rightarrow \pi^{-1}(y)$ anywhere,, – Praphulla Koushik Sep 23 '19 at 16:42
• See Appendix I (the main theorem in the appendix). You apply this theorem locally, to charts covering the given curve in $M$. – Moishe Kohan Sep 23 '19 at 17:18
• I think in the principal bundle case, it is even easier as it is enough to note that $\pi^{-1}(x)$ is a smooth $G$-torsor and the map $\Phi_x$ is $G$-equivariant. – Tobias Diez Sep 24 '19 at 8:27
• Fix a point $x \in X$, then $G \ni g \mapsto x \cdot g \in X$ is a diffeomorphism. This allows you to write your map as a map $G \to G$, which by $G$-equivariance is smooth. In your case, this map is $g \mapsto a g$ where $a \in G$ is fixed. – Tobias Diez Sep 24 '19 at 9:47

As I said in a comment, this holds even for general smooth bundles (which need not be not $$G$$-bundles). You can find a proof for instance in Theorem 9.8, page 80, in:
Assuming that $$\Phi$$ is a vector field on an open subset $$U$$ of $$R^n$$, the Cauchy problem of the form $$x'(t)= \Phi(x), x(t_0)=v\in R^n$$ has unique smooth short-term solutions on relatively compact open subsets in $$U$$. See for instance Appendix I of Kobayashi-Nomizu "Foundations of Differential Geometry".