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For distributions on smooth manifolds one can consider the push-forward which is defined for proper maps, and the pull-back which is defined under certain condition on the wave front set see Hormander The analysis of Linear PD operators I, 8.2.

I need a reference for the following fact:

Let $\phi:X \to Y$ be a proper map of smooth manifolds. Let $Z \subset Y$ be a closed submanifold. Assume that $\phi$ is transversal to $Z$, i.e. for any $x\in X$ s.t. $\phi(x)\in Z$ we have $d_x(T_x(X))+T_{\phi(x)}Z=T_{\phi(x)}Y$. Let $W=\phi^{-1}(Z)$. Note that it is a submanifold. Let $\xi\in C^{-\infty}(X)$ s.t. $WF(\xi) \cap CN_{W}^X \subset X,$ were $WF$ is the wave front set and $CN$ is the co-normal bundle. Then

  1. $WF(\phi_*(\xi))\cap CN_{W}^Y \subset Y.$ (This follows easily from -- Hormander The analysis of Linear PD operators I, 8.2)

  2. $\phi_*(\xi)|_Z=(\phi|_W) _*(\xi|_{W})$. The restrictions are defined because of the conditions on the wave front set.

Thank you very much

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