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Let $(M,m_0)$ be a real analytic manifold. Let $\pi$ be a cotangent at $m_0$. How can one prove that there exists a global analytic function $\theta:M\rightarrow \mathbb{R}$ such that $d\theta(m_0)=\pi$?

I know such a global function $\theta:M\rightarrow \mathbb{R}$ will have to exist because of the paper: "On Levi's Problem and the Imbedding of Real-Analytic Manifolds Hans Grauert" published in Annals of Mathematics in 1958. However, I would like to know if there is an easier way to prove this, since my goal is less than what that paper acheives .I only want to prove the existence of a function rather than get a global embedding in a finite dimensional euclidean space.

Thank you,

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  • $\begingroup$ If $M$ is compact, does not this contradict the Maximum Principle? $\endgroup$ Commented Aug 15 at 7:34
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    $\begingroup$ @FrancescoPolizzi Thank you for your comment. Can you please elaborate ? Please note I do not require $\theta$ to be harmonic $\endgroup$ Commented Aug 15 at 8:49
  • $\begingroup$ Ok, sorry. I was assuming harmonicity. $\endgroup$ Commented Aug 16 at 5:43

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