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The paper gives a characterization of the traces, that are indicated in the title, in the case of a compact manifold M.

Let $\pi :X\to \tilde{X}$ be a morphism of compact differentiable manifolds and $(d,N)$ [resp. $\tilde{d}$] a differential operator on $(X,\partial X)$ [resp. $\tilde{X}$]. Under suitable assumptions the formula: $dim\mathrm{Ker}\tilde{d}-dim\mathrm{Ker}(d,N)=k$ is given, where $k$ is an integer depending only onthe order of $d$ and $N$ and on the number of components of $\partial X$.