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Let $\alpha$ be a differential form of degree $p+1$ on $M\times\mathbb{R}$, where $M$ is an arbitrary smooth manifold, and $p$ a non negative integer.

Can $\alpha$ be always written as $\beta+\gamma\wedge dt$, where $\beta$ and $\gamma$ are differential forms on $M\times\mathbb{R}$ of degree $p+1$ and $p$ respectively, such that $\frac{\partial}{\partial t}\lrcorner\beta=0$ and $\frac{\partial}{\partial t}\lrcorner\gamma=0$? and, in the affirmative case, are $\beta$ and $\gamma$ uniquely determined?

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    Good, this is now correct. My intention was simply to point out that the answer is no if you want $\beta$ and $\gamma$ to be forms on $M$ alone :)2011-04-11

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yes and yes, for $\alpha$ of arbitrary degree. I'll rather write $\alpha=\beta+dt\wedge\gamma$ to save a sign (i.e. I possibly changed the sign of $\gamma$). Then $\gamma=\frac{\partial}{\partial t}\lrcorner \alpha$ and $\beta=\alpha-dt\wedge\gamma$. (you can easily see that these equations must be true if your conditions are satisfied, and also that they give $\beta$ and $\gamma$ satisfying your conditions)

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    Thanks, this is properly what I was searching for. Consequently I have even edit the question to make it, like your answer, independent on the degree of the form α.2011-04-11