Let $E$ be a real analytic vector bundle on an analytic manifold $M$. Assume that $E$, as a smooth vector bundle, is a trivial bundle.



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I need to the answer to this question for the following question:


[Analytic version of the Cartan lemma](https://mathoverflow.net/questions/161823/analytic-version-of-the-cartan-lemma#comment413134_161823)



analytic vector bundles
Here's just a sketch, maybe someone else can fill in the details. Most of the steps will require the use of the nontrivial fact that $C^\omega(M)$ is dense in $C^\infty(M)$. Let's assume $M$ is compact.


Step 1: Construct a real-analytic embedding of vector bundles $E\to M\times\mathbb R^N$ for some finite $N$. This gives a real-analytic classifying map $M\to\operatorname{Gr}\_n(\mathbb R^N)$.


Step 2: Increase $N$ to $N'$ so that the composition $\gamma:M\to\operatorname{Gr}\_n(\mathbb R^N)\to\operatorname{Gr}\_n(\mathbb R^{N'})$ is smoothly null-homotopic.


Step 3: Approximate this smooth null-homotopy of $\gamma$ with a real-analytic homotopy from $\gamma$ to a map $\gamma':M\to\operatorname{Gr}\_n(\mathbb R^{N'})$ which is contained in an $\epsilon$-ball over which the universal bundle is real-analytically trivial.


I suspect this argument also works for $M$ non-compact (taking $N=\infty$) though I haven't checked carefully.


