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I'm sorry, I dont understand yet. How can I use $\varphi_* \alpha = 0$ to calculate $\varphi^* D . \alpha$? I'm aware it must be simple but I do not see it..
You are right, I was trying to make the question general, even though I am interested in the case where $dim(\varphi_* \alpha)< dim( \alpha)$. I will change the question. Can you explain though why the product would be zero in this case? It was my guess but I can't really prove it