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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.

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  • $\begingroup$ 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. $\endgroup$ – Moishe Kohan Sep 23 '19 at 16:13
  • $\begingroup$ @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,, $\endgroup$ – Praphulla Koushik Sep 23 '19 at 16:42
  • $\begingroup$ See Appendix I (the main theorem in the appendix). You apply this theorem locally, to charts covering the given curve in $M$. $\endgroup$ – Moishe Kohan Sep 23 '19 at 17:18
  • $\begingroup$ 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. $\endgroup$ – Tobias Diez Sep 24 '19 at 8:27
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    $\begingroup$ 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. $\endgroup$ – Tobias Diez Sep 24 '19 at 9:47
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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:

I.Kollar, P.Michor, J.Slovak, "Natural operators in differential geometry", Springer-Verlag, 1993.

But the key is a local theorem from ODEs (proof of which you can find in any graduate-level textbook on ODEs):

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".

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  • $\begingroup$ Thank you. I checked Natural operators in differential geometry.. did not understand completely, I might need day or two but I am sure I will get it :) $\endgroup$ – Praphulla Koushik Sep 24 '19 at 2:26

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